The Blues must have thought the early danger had passed when their wall blocked a Dominik Szoboszlai free-kick, with the Hungarian seeking what would have been a spectacular sixth direct free-kick goal of the season.
But the ball found its way out to Ryan Gravenberch, who came up with a brilliant strike of his own, curling it into the top corner from the edge of the box. A great cross from Marc Cucurella almost led to a quick response, but Virgil van Dijk put his head among the boots to repel the danger from within the six-yard box.
The Liverpool captain then had a golden chance to double the advantage at the other end, finding himself all alone at the far post from a Szoboszlai cross, but it was a defender’s finish, looping harmlessly over the crossbar.
Cucurella threatened to make the Reds rue that wastefulness, forcing a couple of smart interventions from Giorgi Mamardashvili, and the mounting pressure eventually told shortly beyond the half-hour mark.
Enzo Fernández whipped in a low free-kick, and it ended up going all the way in, albeit with some debate as to whether it grazed the studs of Wesley Fofana.
The visitors ended the half in the ascendancy, and Fernández should have put his goal-scorer status beyond all doubt before the break, with only a good save keeping things level.
Chelsea looked to have claimed the lead within minutes of the restart, the threat once again coming down Liverpool’s right side. But Cole Palmer’s goal was chalked off after a VAR review, with Cucurella adjudged to have been leaning marginally offside in the build-up.
Then it was the hosts’ turn to be denied by the flag, with Cody Gakpo straying slightly beyond the last man before heading it down for Curtis Jones to convert.
Arne Slot invoked rare boos from Anfield when he withdrew Rio Ngumoha midway through the half, replacing him with record signing Alexander Isak.
Despite the disquiet, Liverpool did begin to take control of the contest again, and only the post denied Szoboszlai the go-ahead goal, before the bar repelled Van Dijk. But ultimately, neither side could find the winner, leaving a dissatisfying result all round.
Chelsea are just four points clear of 13th, which would mark their worst-ever PL finish, while the Reds will have to wait another week to confirm UCL football unless Bournemouth lose at Fulham.