Thursday's draw left the Blues four points adrift of table-toppers Manchester City.
Chelsea, missing their first-choice strikers due to COVID-19, failed to turn early dominance into goals before Mason Mount made the breakthrough in the 70th minute when he took a pass from Reece James and fired past Jordan Pickford.
Mount had missed one of the hosts' many chances in the opening period when, clean through, he was denied by Pickford's outstretched foot.
The visitors drew level within four minutes of Chelsea going ahead when Anthony Gordon floated a looping free-kick to the far post and 19-year-old Jarrad Branthwaite stretched out a boot to turn the ball past goalkeeper Edouard Mendy.
Chelsea struggled to make further inroads without Romelu Lukaku, Timo Werner and Callum Hudson-Odoi, who had all tested positive for COVID-19. Kai Havertz was also unwell and awaiting the results of a test.
The draw left Chelsea in third place behind Manchester City who thrashed Leeds United 7-0 on Tuesday, and Liverpool.
Chelsea coach Thomas Tuchel said the result was "hard to swallow" but added he would not use the absences as an excuse for the failure to keep pace at the top of the table.
"I think we missed big chances in the first half to get things straight and in the second half we did not find the rhythm consistently," he told BT Sport. "We gave it away with one free kick."
Up north, the Reds' Trent Alexander-Arnold scored a spectacular goal as they bounced back from the loss of three players to suspected positive COVID-19 tests to beat Newcastle.
Liverpool moved onto 40 points from 17 games, one behind defending champions City.
With Virgil Van Dijk, Fabinho and Curtis Jones all ruled out for the hosts, Jonjo Shelvey struck the opener for Newcastle against the run of play in the seventh minute, with a swerving, dipping drive past unsighted goalkeeper Allison Becker.
The lead lasted 14 minutes before Diogo Jota fired home the rebound after Martin Dubravka saved his initial header to level, with Newcastle's players complaining the ball should have been put out of play as Isaac Hayden was down injured in his own box.
Mohamed Salah put Liverpool ahead four minutes later and Alexander-Arnold got the third with an 87th-minute rocket to wrap up a win that leaves Newcastle second from bottom on 10 points.
"I've been waiting for that for five years, waiting to hit it clean like that," Alexander-Arnold told BT Sport. "I've had a few from the edge of the box ... I've caught that one sweet and it's just nestled in the top corner."
The England fullback said he and his teammates had done their best to ignore the possible positive COVID tests at the club as the games come thick and fast over Christmas.
"We're focused on our football until we're told otherwise. We'll be in there every day, recovery again tomorrow and make sure we're fit for Sunday," he said.
"We've got a big game, going down to Tottenham - it'll be an interesting one."
