The first half provided few clear-cut chances for either side, though it was Paul Clement's men who dominated possession against a Hammers outfit which failed to get out of second gear.
Referee Roger East blew the half-time whistle on a chorus of boos from the frustrated home fans and, although the second half saw a better display from their team, they still flattered to deceive.
However, a match devoid of thrills for the most part found a moment of magic late on, with substitute Arthur Masuaku slicing through the Swansea defence before setting up Sakho to finish from close range.
Both sides threatened around the quarter-hour mark, with Andy Carroll sending a volley bouncing off the turf and into Fabianski's hands, while a minute later, Wilfried Bony's ferocious effort forced a smart save from Joe Hart.
That was as exciting as it got in terms of first-half action, with little more to show from either side, and the half-time whistle saw the game still on a knife edge.
Following the restart, there was yet more frustration on display from the West Ham fans as Winston Reid clumsily sliced a clearance behind to gift the opposition a corner.
To their credit, however, the hosts improved and tested Swansea's defence, with a great cross on 52 minutes from Antonio flashing across goal and needing the slightest of nudges from Carroll to break the deadlock.
On 75 minutes, the Swans came close to opening the scoring when Martin Olsson cut in off the left flank and tried to curl an effort into the bottom corner, but his strike bounced past the post.
Aside from Olsson's chance, the visitors' struggles in attack were plain for all to see and, late on, their impressive defensive record on the road gave way.
Just four minutes after Carroll smashed an effort against the woodwork, Masuaku burst down the left, ghosted beyond Kyle Naughton and whipped the ball across the goalmouth for Sakho to tap beyond Fabianski.
The result sees West Ham climb to 15th in the Premier League table, while Swansea remain in the bottom three on goal difference.