The Blues saw fourth-placed Tottenham Hotspur slip up away to relegation-threatened West Bromwich Albion on Saturday, and they took full advantage as Olivier Giroud's first-half header handed them three vital points in West London.
Liverpool - who were playing their first match since booking their place in the Champions League final - knew that a drew would have been enough to effectively wrap up another campaign in Europe's top-tier competition next season, but they now go into their final game against Brighton & Hove Albion needing a win to guarantee their place in the top four.
Chelsea went into the match with Antonio Conte's rallying cry for maximum points from their final three games ringing in their ears, but it was Liverpool who made the faster start and they broke in behind the hosts' defence as early as the third minute when Roberto Firmino latched on to Trent Alexander-Arnold's pass and saw his volley from a tight angle parried away by Thibaut Courtois.
The home side lacked the urgency that many expected of them in the opening stages, and they wasted a promising position 14 minutes in when Giroud took the ball off the toes of Tiemoue Bakayoko when his compatriot was much better placed to meet Victor Moses's cutback.
Chelsea did have their first shot on target just two minutes later, though, when a mistake from Alexander-Arnold - again deployed in a midfield role - allowed them to come forward down the left flank, only for Marcos Alonso to fire his 20-yard effort straight at Loris Karius.
It was Liverpool who were on top for the opening half-hour, though, and Sadio Mane squandered the first of his chances after 21 minutes when he scooped Nathaniel Clyne's low cross over the bar after the ball had arrived behind him.
Mane threatened again midway through the half when he burst forward from midfield before letting fly, but Courtois was equal to his effort and then got a crucial touch to the resulting corner to take it away from the head of Virgil van Dijk.
Mane continued to look the most likely to break the deadlock for Liverpool, but Courtois once again came out on top in their duel on the half-hour mark when he pushed another curling strike away before collecting a tamer follow-up effort after Firmino had kept the move alive.
Chelsea responded by going straight up the other end and creating their best chance of the match, and Bakayoko's glancing header which dropped narrowly wide of the far post proved to be a warning shot of what was to come just minutes later.
The hosts' opener came against the run of play as Moses advanced down the right flank before cutting inside and putting a cross in for Giroud, who got in front of his man to flick his header into the far corner, leaving Karius rooted to the spot.
Suddenly it was Liverpool on the back foot, and Chelsea came agonisingly close to a second moments later when Cesc Fabregas broke down the right channel before delivering a teasing low cross into the box which only needed a touch.
Mohamed Salah - still chasing a 32nd league goal of the campaign which would set a new 38-game scoring record for the Premier League - struggled to get into the game for the most part and had a moment to forget late in the first half when he was correctly booked for diving just outside the penalty area.
The PFA and FWA Footballer of the Year did have his first half-chance just 41 seconds after the interval when he linked up well with Firmino on the edge of the box, but his resulting effort was tame and easy for Courtois to gather.
It was a sign of things to come as the second half got off to a quick start, with Moses flashing a dangerous low ball across the face of goal before Alexander-Arnold and N'Golo Kante both failed to hit the target from range at either end of the field.
Eden Hazard's dancing feet took him to the byline down the left channel 11 minutes into the second half, but his powerful drive across the face of goal was met by an unconvincing touch from Karius before Andrew Robertson was able to smuggle it behind at the back post.
The hosts did have the ball in the back of the net from the resulting corner when Antonio Rudiger turned Gary Cahill's header home from close range, but the offside flag cut Chelsea's celebrations short.
Liverpool were struggling to find a way through Chelsea, though, and other than a tame Bakayoko effort from range neither side came close again until Alonso produced a stunning Marco van Basten-esque volley which flashed narrowly wide of the far post with Karius beaten.
The visitors began to dominate again in the final 10 minutes as they pushed for a late equaliser which would all but secure their top-four place with a game to spare, and it almost came from an unlikely route when Cesar Azpilicueta's botched clearance from the edge of his own six-yard box was gratefully collected by Courtois.
The best chances any Liverpool player had in the closing stages both fell to former Chelsea man Dominic Solanke, but the striker missed the target with two headers, sending both over the crossbar.
One final Liverpool attack saw Van Dijk nod the ball into a dangerous area, but Courtois was there to gather it as Chelsea held on for a vital victory which ends a six-match winless run against the Reds and could take the top-four race down to the final day of the season.
Liverpool, meanwhile, have now failed to score in two successive Premier League games for the first time since August 2015 and could be left needing victory on the final day to secure their place in the top four.
CHELSEA (3-5-1-1): Courtois; Azpilicueta, Cahill, Rudiger; Moses (Zappacosta 89'), Fabregas (Pedro 93'), Bakayoko, Kante, Alonso; Hazard (Willian 86'); Giroud
LIVERPOOL (4-3-3): Karius; Clyne (Henderson 59'), Lovren, Van Dijk, Robertson (Solanke 74'); Alexander-Arnold, Wijnaldum, Milner (Moreno 89'); Salah, Firmino, Mane