The visitors took the lead after just four minutes when Cesc Fabregas curled his effort into the top corner, compounding a miserable day for Swansea after the three teams below them in the table had all picked up points.
Chelsea failed to build on their good start, but the hosts could not respond despite some late pressure as they were left with just a one-point cushion above the relegation zone.
Find out how all of the action unfolded courtesy of Sports Mole's minute-by-minute updates below.
This promises to be a huge match at both ends of the table, with relegation-threatened Swansea having seen results go against them so far today and Champions League-chasing Chelsea needing to avoid any further slip-ups in the closing stages of the season.
We will take a close look at the plight of both teams a little later, but first let's check out the team news...
Well, the headline news as far as Chelsea are concerned is that Alvaro Morata is once again left out in favour of Giroud, who was also preferred to lead the line for the FA Cup semi-final win over Southampton last weekend.
Morata's form over the course of the season has been questioned on numerous occasions, but he actually has three goals in his last six appearances for the Blues, which is as many as he managed in his previous 22 before that, so there are certainly signs of improvement. That is not enough to earn him a starting berth today, though, as he is forced to settle for a place amongst the subs.
Giroud has been most effective as an impact player so far this season, though, with all six of his Premier League goals in 2017-18 coming off the bench - a Premier League record for a single campaign. The Frenchman does have a decent record at this stadium, though, having scored two and set up two more in his four previous league starts here.
Hazard is one of a number of Chelsea players who has struggled to produce his best form on a consistent basis this season, but he will be hoping for a strong end to the campaign ahead of a World Cup which promises so much for his Belgian side.
One player who does make another start is Emerson, and this may well be his final chance to impress Conte for a while as Marcos Alonso is serving the final game of his three-match ban, and will be expected to regain his starting spot as soon as he is back - unless Emerson can pull something special out of the bag today.
It was thought that Carvalhal might switch to three at the back, but instead it looks as though he has moved Naughton into a more central role as one of the three centre-backs.
The defensive is completed by Van der Hoorn and Alfie Mawson, both of whom also started against Manchester City last time out but were powerless to prevent the newly-crowned champions from running riot in a 5-0 win.
We may have to wait a while to see whether he will extend that run, though, as all 15 of his previous Premier League goals have been scored in the second half.
On-loan midfielder Renato Sanches, who arrived in South Wales with big expectations which he has struggled to live up to so far, returns to the squad for the first time since the end of January following a hamstring problem.
Stoke and Southampton are the two of the biggest importance to the Swans, and after the Potters picked up an unexpected point at Anfield today, Southampton are now leading Bournemouth in the closing stages of their match. That latter result would leave Swansea just one point clear of danger.
It is Southampton who pose the biggest threat right now as they have played the same number of games as Swansea and are just one point behind, whereas Stoke and West Brom are further back having played more games. Incidentally, despite West Brom winning today, a win for Swansea here would relegate the Baggies.
After today's match Swansea have three games remaining, the first of which comes against Bournemouth. Then, though, they must face 18th-placed Southampton and 19th-placed Stoke in their last two games, and defeat in either could prove fatal for their survival hopes.
Indeed, the Swans have only lost one of their last nine home games across all competitions, including a run of seven wins in a row prior to defeat and a draw in their last two. That run also includes victories over Liverpool and Arsenal, which should give them plenty of confidence heading into this match.
Swansea's improvement in front of their own fans has been substantial, winning just two of their opening 11 home league games but now picking up 13 points from the last 15 on offer in South Wales.
Their three defeats in that time having come against Tottenham and the two Manchester clubs, though, while during an ongoing five-match winless streak in the Premier League, four of those games were away from home, so Carvalhal won't be sounding the panic alarms just yet.
Carvalhal would have certainly taken this position had you offered it to him when he took over with the club bottom of the table, and a victory today would see Swansea become the first non top-six side to beat three of those teams this season.
Liverpool's draw with Stoke earlier in the day saw them drop more points in the top-four race, but it will be Tottenham who are most firmly in Chelsea's sights at the moment. Five points currently separate the two London clubs either side of that all-important dotted line, so there is still plenty of work for Chelsea to do.
Spurs will also be looking to bounce back from seeing their final hope of silverware go begging once again in the FA Cup semi-finals last weekend, while Chelsea are riding the crest of a wave having booked their place in the Wembley showdown against Tottenham's conquerors Manchester United.
Last season's champions come into this match having won each of their last three games across all competitions, having only won two of their previous eight prior to that run, losing four. Victory today would see them string together four wins in a row for only the second time this season, and the first time since September.
However, you have to go back to November for the last time Chelsea won three Premier League games on the bounce, which is a surprisingly poor record for a team that won the title so convincingly last season.
The Blues have failed to keep a clean sheet in their last seven Premier League outings and have only managed one in their last 10, conceding 18 goals in that time. It is a far cry from the solid unit which saw them storm to the title last season, and something Conte needs to address heading into the final four games of the season against Swansea, Liverpool, Huddersfield and Newcastle.
Indeed, Chelsea had not won successive away games since November - and had only won two of 12 - prior to this latest run. Another victory here would see them win three away league games in a row for the first time since their opening three of the season.
Should he reach the milestone today then he would become the third-quickest manager to amass 50 wins in the Premier League, behind only Jose Mourinho (63 games) and Pep Guardiola (69 games), and yet still speculation swirls over his future.
This is a difficult one to call considering Swansea's recent record at the Liberty Stadium, and it wouldn't be a major surprise to see them come away from this game with at least a point. Chelsea's need to win may just push them over the edge, though, so in a match which could go either way, we're going to side with the visitors.
SPORTS MOLE SAYS: Swansea 1-2 Chelsea
That victory did come only two years ago, but aside from that they have drawn four and lost eight - including the reverse fixture in November when Antonio Rudiger scored the only goal of the game.
Indeed, the Blues needed a late Diego Costa goal to rescue a draw here in their title-winning campaign last season, with the Spain international completing his brace to canceling out quickfire goals from Sigurdsson and Leroy Fer.
Carlos Carvalhal described this fixture as a "David against Goliath" showdown prior to kickoff and, while that may have contained a touch of hyperbole, it would certainly upset the formbook if the Swans came away with a win here.
SWANSEA STARTING XI: Fabianski; Roberts, Naughton, Van der Hoorn, Mawson, Olsson; Ki, King, Clucas, A Ayew; J Ayew
CHELSEA STARTING XI: Courtois; Azpilicueta, Cahill, Rudiger; Moses, Bakayoko, Kante, Fabregas, Emerson; Hazard, Giroud
Swansea give the ball away in midfield Kante immediately finds Hazard, who breaks away from one attempt to bring him down before putting it in the path of Fabregas. Fabregas takes it on first time and just sweeps his finish beyond Fabianski with a lovely left-footed finish. That's 50 Premier League goals for Fabregas!
50 - Cesc Fabregas has scored his 50th Premier League goal, becoming the third Spanish player to do so in the competition (after Fernando Torres and Diego Costa). Milestone. pic.twitter.com/P3FrLhaW6Y— OptaJoe (@OptaJoe) April 28, 2018
There have not been a lot of chances at either end of the table, but Swansea in particular have struggled to pose any threat in the final third. Chelsea have dominated possession, but they have also failed to turn that into a regular stream of chances.
Hazard then broke away and poked the ball through for Fabregas, who produced a lovely left-footed finish into the top corner to bring up his 50th Premier League goal.
The closest we have come to a second goal actually came via the boot of Alfie Mawson, who almost scored an own goal when he lifted the ball over Fabianski and against his own crossbar on its way over the top.
It has been a poor day for the Swans so far with all three of the teams below them in the table picking up points in the earlier kickoffs.
Survival is still ultimately in Swansea's hands, but today's results have left them just one points clear of the bottom three with three games of the season remaining. Chelsea, meanwhile, close the gap on the top four to just two points courtesy of Cesc Fabregas's 50th Premier League goal, but that could grow against when Spurs are in action on Monday night.
Thank you very much for joining Sports Mole for tonight's Premier League clash between Swansea City and Chelsea as the visitors hang on for a big result at both end of the table. I will leave you with our match report, and be sure to stick around for reaction too.
From me, though, it is goodbye for now!