The Reds – with Virgil Van Dijk and Fabinho back in the side after recovering from Covid-19 – were awarded a penalty in the 16th minute when Mohamed Salah was brought down by Wilfred Ndidi.
However, the Egyptian forward saw his spot-kick saved by Kasper Schmeichel and he then headed the rebound against the crossbar.
Sadio Mane fired over early in the second half, before Leicester took the lead on the hour through substitute Ademola Lookman after the former Everton player was played in by Kiernan Dewsbury-Hall.
Schmeichel saved from Van Dijk at close range late on as the Reds pressed for an equaliser, but the Foxes held out as Brendan Rodgers celebrated a memorable victory over his old club.
Liverpool, held to a draw by Tottenham in their last match, now sit six points off the top, while Leicester are ninth.
Earlier, Crystal Palace had inflicted a fifth straight defeat on bottom club Norwich with a comfortable 3-0 win at Selhurst Park.
The Eagles, missing midfielder Conor Gallagher and with Wilfried Zaha suspended, took an early lead from the penalty spot through Odsonne Edouard after Kenny McLean was adjudged to have tripped Will Hughes.
Norwich had a chance to equalise at a corner, but Jacob Sorensen missed his kick from close range – and Palace broke quickly to make it 2-0 in the 38th minute when Jean-Philippe Mateta drilled the ball into the far corner.
Jeffrey Schlupp fired in a third from a tight angle just before half-time to leave the struggling Canaries crestfallen.
Norwich – beaten 5-0 at home by Arsenal on Boxing Day – improved in the second half, during which 18-year-old Jonathan Rowe was sent on for his first senior club appearance, but could not find a consolation and have failed to score a goal in December.
Palace saw a late fourth from substitute Christian Benteke ruled out for offside, and sit 10th.
West Ham climbed to fifth after returning to winning ways with a 4-1 victory at strugglers Watford.
Emmanuel Dennis had given the Hornets an early lead, but Tomas Soucek equalised after being played into the penalty area on 27 minutes.
Said Benrahma then completed the turnaround just a couple of minutes later with a deflected effort from Michail Antonio’s cross.
West Ham were awarded a penalty in the 58th minute, given after referee Darren England had viewed the pitchside monitor, and captain Mark Noble made no mistake from the spot.
Substitute Nikola Vlasic added a fourth in stoppage time, scoring his first goal for the club, to leave Watford 17th, just two points above the drop zone.
Tottenham missed the chance to close up on the Champions League places after being held to a 1-1 draw at Southampton, who were down to 10 men at the end of the first half.
After Spurs defender Ben Davies saw his effort following a free-kick ruled out for offside, James Ward-Prowse fired past Hugo Lloris to put the hosts ahead in the 25th minute.
Tottenham were level shortly before half-time when Son Heung-min was brought down by Mohammed Salisu, who was sent off for a second caution, and Harry Kane despatched the resulting penalty.
England captain Kane had the ball in the net again soon after the restart, but his goal was ruled out for offside following a VAR review.
Tottenham drop to sixth, but have games in hand on the teams above them.