The Reds fell behind after less than three minutes as Jamie Vardy continued his remarkable record against the Merseysiders, and despite creating a string of chances the hosts could not restore parity before half time.
However, a second-half brace from Mohamed Salah saw Jurgen Klopp's side come from behind to pick up a valuable victory, lifting them four points clear of Arsenal and Tottenham Hotspur in the race for a top-four spot.
Liverpool went into the match boasting the best home defensive record in the Premier League, but it took less than three minutes for their reasoning behind spending a world-record sum on Virgil van Dijk to be reinforced.
Van Dijk was watching on from the directors' box as Joel Matip gifted possession to Leicester in a dangerous area, allowing Vicente Iborra to release Riyad Mahrez who in turn set it up on a plate for Vardy to tuck home an easy finish into an empty net.
It was Vardy's seventh goal in his last five Premier League appearances against Liverpool, making them his favourite opponent in the top flight and giving Leicester a perfect start to the contest.
Liverpool quickly roared back, though, and were soon creating chances almost at will in their search for an equaliser, with Andrew Robertson sending one tantalising cross into the box before Salah somehow placed his finish wide of the target after he had been found by Sadio Mane inside the area.
Salah had another clear chance shortly afterwards when he latched on to James Milner's through-ball, but he took too long over his finish which allowed Harry Maguire to get back and make an important sliding challenge.
It was one-way traffic in Liverpool's favour and Philippe Coutinho was the next to test Kasper Schmeichel before the Dane reacted quickly to thwart Milner after the ball had broken to him inside the area.
The hosts did finally have the ball in the back of the net after 19 minutes when Mane converted from Robertson's cross, but the linesman's flag correctly cut Liverpool's celebrations short.
Unusually it was Salah who was guilty of wasting Liverpool's best chances of the first half, and the Egyptian winger again failed to hit the target shortly afterwards when he only had the keeper to beat following a one-two with Emre Can on the edge of the box.
Liverpool refused to relent in their search for an equaliser, though, and Roberto Firmino was the next to come close when he drew a smart low stop from Schmeichel.
Joe Gomez and Firmino threatened again before the half-hour mark, but Leicester did eventually stem the tide of Liverpool chances and managed to reach half time with their one-goal advantage still somehow intact.
The hosts quickly picked up where they left off at the start of the second half, though, and within seven minutes of the restart they had their equaliser as Salah atoned for his own missed chances in the first half.
Mane's back-heel released the winger, who delayed his finish to draw in two tackles before drilling the ball past Schmeichel for his 22nd goal of the season.
Salah almost added to that tally again just before the hour mark when he raced onto a Coutinho pass down the right channel, but he lifted his finish over Schmeichel and onto the roof of the net.
Mane had the ball in the back of the net for the second time with just under 25 minutes of the match remaining, but once again it was correctly ruled out for offside after he had strayed marginally too early to get onto Coutinho's exquisite pass.
Leicester provided a rare threat of their own moments later when the ball was only cleared as far as Wilfred Ndidi on the edge of the area and the midfielder fired a well-struck half-volley inches wide of the target.
Liverpool did finally take the lead with just under 15 minutes remaining, and again it was Salah who got the goal as he span away from Maguire's tight marking before tucking his finish past Schmeichel with aplomb.
Leicester's pressure began to grow in the closing stages as they went in search of a route back into the match, but Liverpool held out to extend their unbeaten run to 15 matches and steal a march on Arsenal and Tottenham, neither of whom are in action today.
The Reds have now scored a top-flight club-record 77 goals from their opening 30 matches across all competitions so far this season - a tally only Paris Saint-Germain can beat from Europe's top five leagues.
Leicester, meanwhile, have now taken only one point from the last 12 on offer in the Premier League and have lost back-to-back away league games for the first time since their opening two of the season - although defeat for Everton means that they remain in eighth place.
Liverpool (4-3-3): Karius; Gomez, Matip, Lovren, Robertson; Coutinho (Klavan, 89'), Can, Milner; Salah (Wijnaldum, 83'), Firmino, Mane (Oxlade-Chamberlain, 72')
Leicester (4-4-1-1): Schmeichel; Amartey, Morgan, Maguire, Fuchs; Mahrez (Okazaki, 73'), Ndidi, Iborra, Albrighton (James, 83'); Gray; Vardy (Slimani, 74')