Jurgen Klopp’s men came into the game knowing rivals City had won 3-1 at Crystal Palace earlier on Sunday so they needed the win to regain their place at the summit.
However, despite boasting a 37-game unbeaten run on home soil, the Reds had not won any of their last six Premier League games at Anfield against Chelsea, who also needed the win for their top-four ambitions.
After a goalless opening first half, Liverpool sent the home fans delirious with two quick goals early in the second period with a Sadio Mane header followed by a sumptuous strike from Mohamed Salah.
That opened the game up with the hosts looking to add more goals to their tally, while Eden Hazard hit the post and spurned another great chance for the visitors in a frantic 45 minutes.
However, neither side could find a way through again and Liverpool move two points above City on 85 points having played a game more, while Chelsea remain in fourth on 66 points.
Andy Robertson replaced James Milner and Joel Matip came in for Dejan Lovren in just two changes to the Liverpool side which defeated Porto 2-0 in the Champions League in midweek.
The Chelsea side boasted six changes from the side which beat Slavia Prague 1-0 in the Europa League on Thursday as David Luiz, Emerson, N'Golo Kante, Ruben Loftus-Cheek, Eden Hazard and Callum Hudson-Odoi replaced Andreas Christensen, Marcos Alonso, Ross Barkley, Mateo Kovacic, Olivier Giroud and Pedro.
Liverpool started with purpose and had the first effort of note in the sixth minute when Mane’s cross from the left dropped over Jordan Henderson but fell kindly to Salah who saw his left-foot volley held by Kepa Arrizabalaga.
Naby Keita picked up possession from Trent Alexander-Arnold and curled a strike into the hands of Kepa, before Hazard tested Reds goalkeeper Alisson at the other end on 20 minutes with an effort from the left of the area after being played in by Willian.
Chelsea’s influence on the game continued to grow as they continued to stifle the hosts and hit them on the counter-attack with Hazard adopting a ‘false nine’ role in the absence of any recognised strikers.
However, it was Liverpool who again went close on 26 minutes when Henderson's effort from the edge of the area took a deflection off Luiz and lopped just over the crossbar.
Willian was then allowed to run towards the Liverpool goal from the halfway line, before he curled one just wide of Alisson's far corner from the left channel.
Back came the Reds on 39 minutes with Salah the provider from the left side of the area and his pass picked out Mane who hit his effort wide of the near post.
There was some hope for Liverpool a minute later when the Chelsea backline was disrupted by Antonio Rudiger being forced off because of an injury with Christensen replacing him.
And, they went close right on half-time when Matip got a touch on Alexander-Arnold's inswinging free-kick from wide out on the left right in front of goal, only for Kepa to save with his feet. However, the offside flag was up on the far side so it wouldn’t have counted anyway.
Liverpool started the second half brightly and the game swung firmly in their favour with two goals in as many minutes.
On 51 minutes, Salah's charge into the right side area resulted in the ball rolling to Henderson and his chip to the back post was met by Mane who had the simple task of guiding his header into the net from close range.
The Reds' fans had barely stopped celebrating that goal when Salah picked up possession to the right of centre and slammed a brilliant 25-yard strike beyond the dive of Kepa and into the far top corner of the net to send them into raptures.
Liverpool's tails were well and truly up having secured a two-goal cushion and they almost added a third when Roberto Firmino hit a low strike inches wide of the far post on 57 minutes.
However, Chelsea responded and they should have got a foothold in the game just before the hour mark when Hazard got in behind the Reds' backline to latch on to Christensen's ball over the top, only to see his effort rebound off the post with Alisson beaten.
Hazard spurned another good chance a minute later when he met Willian’s clipped ball into the box but then scuffed his shot to Alisson.
Salah looked to find the far corner again on 66 minutes, but this time Kepa was equal to the task as he pushed the ball wide, before Alexander-Arnold was denied by the Blues' custodian six minutes later.
Mane almost grabbed his second and Liverpool’s third in the 89th minute when he surged past Christensen down the left side of the area and lashed in an angled drive which Kepa could push over his own crossbar.