The draw meant Chelsea remained second on 43 points from 21 games, one point ahead of Liverpool, who have a game in hand, and leaders Manchester City who have 53 points from 21.
With both sides weakened by key absences - and Chelsea dropping striker Romelu Lukaku after he criticised the tactics of coach Thomas Tuchel - Liverpool quickly seemed to have the game in their grip after goals from Sadio Mane and Mohamed Salah before they depart for the Africa Cup of Nations.
Mane capitalised on an error by Trevoh Chalobah to open the scoring in the ninth minute and Salah doubled the lead 17 minutes later when he ghosted past Marcos Alonso to lift the ball over goalkeeper Edouard Mendy from a tight angle.
But the Blues clawed their way back into the game in the 42nd minute when Mateo Kovacic majestically volleyed in off a post from the edge of the box after a punched clearance by Liverpool reserve keeper Caoimhin Kelleher.
Four minutes later, Chelsea were level when Christian Pulisic latched onto a pass by N'Golo Kante that beat Liverpool's offside trap and the American forward coolly lifted the ball over Kelleher into the net.
Both sides went close in the second half with Mendy and Kelleher pulling off good saves.
Late goals from Stuart Dallas and substitute Daniel James earned Leeds United a crucial 3-1 victory over Burnley, a win that moved the home side eight points clear of the relegation zone.
Leeds had not played since Dec. 18 prior to Burnley's visit due to coronavirus-related postponements, and that additional break allowed them to get some much-needed key players back from injury.
Burnley remain third from bottom.
Brighton & Hove Albion earned a hard-fought 3-2 victory at Everton, with Alexis Mac Allister scoring twice and Dan Burn grabbing a third.
Brighton move up to eighth with Everton 15th.
Mads Roerslev scored his first goal for Brentford to give them a comeback 2-1 home win over Aston Villa after Yoane Wissa had cancelled out an early Danny Ings opener for the visitors.
It was a memorable afternoon for the 22-year-old Dane, who joined Brentford in 2019, as he also provided the assist for Wissa's equaliser.
The result lifted Brentford into 12th place on 23 points from 19 games, one point ahead of 13th-placed Villa.