Mohamed Salah saw his first-half penalty cancelled out by Nicolas Jackson, before Curtis Jones scored what proved to be a 51st-minute winner.
The narrow win has moved Liverpool back to the top of the table, while it has left Chelsea seven points adrift in sixth position.
Here, Sports Mole provides player ratings from Liverpool's successful home outing against Enzo Maresca's side.
Liverpool player ratings vs. Chelsea: Jones, Salah play decisive roles in home victory
GOALKEEPER
Caoimhin Kelleher - 7/10
Only had to make one save during the contest, but he ultimately did everything he needed to in Alisson Becker's absence.
DEFENCE
Trent Alexander-Arnold - 7/10
Struggled to have a real impact in attacking areas. Made a number of tackles and helped Liverpool's defensive performance. Might have been nervous following his first-half challenge on Sancho in the box which was ultimately waved off by the referee.
Ibrahima Konate - 7/10
Produced a resolute display in the centre of the Liverpool defence. Picked up a late yellow for a cynical foul on Christopher Nkunku.
Virgil van Dijk - 7/10
Was not tested as much as he may have expected. Nevertheless, he did not put a foot wrong and played his part in a crucial victory.
Andrew Robertson - 5/10
Found it tough going against Noni Madueke, especially in the first half. However, he did make an important block in the closing stages.
MIDFIELD
Curtis Jones - 9/10
Made a superb block to stop Cole Palmer from opening the scoring. The Liverpool midfielder then won the penalty which allowed Salah to break the deadlock. Thought he had won another first-half penalty before it was overturned on review. Ultimately produced the game's defining moment with his second-half goal.
Ryan Gravenberch - 7/10
One of his quieter performances of the season. Still produced a solid display and kept things ticking over in the Liverpool midfield.
Dominik Szoboszlai - 5/10
Probably should have doubled Liverpool's advantage with a golden first-half chance. Apart from that, he struggled to offer to Liverpool's attacking work.
ATTACK
Mohamed Salah - 8/10
Showed great composure to dispatch a first-half penalty, especially after Jackson made an attempt to put him off before the spot-kick. Provided a inch-perfect cross to tee Jones up for the winning goal.
Cody Gakpo - 6/10
Denied a first-half goal by the assistant's race flag. That ultimately proved to be his best moment of an underwhelming performance before he was replaced in the 66th minute.
Diogo Jota - 6/10
Picked up a knock from an early challenge with Tosin Adarabioyo. Attempted to play on but ultimately had to be withdrawn on the half-hour mark.
SUBSTITUTES
Darwin Nunez - 7/10
Replaced the injured Jota in the first half. Picked up a yellow card for a foul on Badiashile. Won a number of fouls and nine duels in a battling performance at Anfield.
Luis Diaz - 6/10
Offered a threat as the game opened up in the closing stages. However, was unable to take full advantage of his two shots, including one effort on target.
Alexis Mac Allister - N/A/10
Replaced Jones in the 81st minute.
Joe Gomez - N/A/10
Came on for Alexander-Arnold in the 81st minute. Made some useful interventions to help Liverpool get over the line.
Chelsea player ratings vs. Liverpool: Jackson makes useful contribution as Cole Palmer fails to deliver
GOALKEEPER
Robert Sanchez - 5/10
Nearly got caught out with a loose pass in the first half. Would have been relieved to see a penalty decision overturned following his collision with Jones. Was unable to stop Salah from the penalty spot, but should have been more committed for Liverpool's second goal.
DEFENCE
Reece James - 7/10
Was given a surprise start in Maresca's defensive shake-up due to the suspensions of Marc Cucurella and Wesley Fofana. Unsurprisingly, he was withdrawn in the second half, with Chelsea reluctant to take too many risks on his return to action.
Tosin Adarabioyo - 5/10
Handed his second Premier League start as a Chelsea player. Received an early yellow card for a clumsy challenge on Jota. That early caution may have been a factor in Maresca's decision to withdraw him in the 53rd minute.
Levi Colwill - 5/10
Would have been frustrated with his rash challenge which led to Liverpool's penalty. However, he did complete more passes than any other player on the pitch.
Malo Gusto - 7/10
Showed great versatility to start as a left-back before switching over to his familiar right side in the second half. Was useful at both ends of the pitch before he saw a late shot blocked.
MIDFIELD
Moises Caicedo - 7/10
Produced a brilliant through ball for Jackson to pull Chelsea level. Looked to have struck up a strong partnership with Lavia, before his teammate was withdrawn in the second half.
Romeo Lavia - 7/10
Made a positive impression in the middle of the park. Achieved a 96% pass accuracy and completed all three of his dribbles during the contest. Would have been disappointed to be withdrawn in the early stages of the second period.
ATTACK
Noni Madueke - 7/10
Caused Robertson a number of problems in the first period. While he was one of Chelsea's main attacking outlets, his threat wained as the game went on.
Cole Palmer - 4/10
A forgettable display from the Chelsea talisman. Had a shot blocked by Jones and had a couple of other chances. However, he spent the majority of the contest on the periphery.
Jadon Sancho - 6/10
Had a few promising touches and had a small shout for a penalty following a challenge from Alexander-Arnold. However, his game was cut short after being replaced at the break.
Nicolas Jackson - 8/10
Should have done better with a promising chance in the first period, but he showed great composure to beat Kelleher in the opening stages of the second half.
SUBSTITUTES
Pedro Neto - 7/10
Introduced as a half-time substitute. Made a positive impact with his driving runs causing the Liverpool defence a number of problems. Certainly staked his claim to be in the starting lineup.
Enzo Fernandez - 5/10
Was unable to match the heights of the man he replaced as he failed to make the desired impact.
Renato Veiga - 5/10
Collected a yellow card for a cynical foul on Diaz and looked shaky at times in his substitute appearance.
Benoit Badiashile - 7/10
Was involved in an intriguing battle with Nunez. Made several important interventions and looked more assured than Adarabioyo.
Christopher Nkunku - 6/10
Threatened to score with a late chance which hit the post.