The draw leaves Liverpool fourth in the table, edging closer to Champions League qualification, while Chelsea remain ninth after earning a point under interim manager Calum McFarlane.
Liverpool made a bright start and took the lead through Gravenberch, who produced a superb curling finish from the edge of the penalty area after being teed up by 17-year-old Rio Ngumoha. The hosts looked in control early on, but wasted chances allowed Chelsea back into the contest.
Virgil van Dijk spurned a golden opportunity to double the lead, firing over from close range, and Chelsea capitalised on Liverpool's lack of ruthlessness. Enzo Fernandez levelled before half-time when his low free-kick from the right drifted untouched into the far corner, with confusion in the box involving Wesley Fofana and Liverpool goalkeeper Giorgi Mamardashvili.
Chelsea grew into the game and could have gone ahead before the interval, with Fernandez again threatening, but Liverpool survived to reach half-time level.
The visitors briefly thought they had turned the match around early in the second half when Cole Palmer found the net, but the goal was disallowed after Marc Cucurella was ruled offside in the build-up.
Liverpool also saw a goal ruled out, as Curtis Jones' close-range header was chalked off after Cody Gakpo was adjudged offside in the build-up.
Chances came at both ends as the match opened up. Dominik Szoboszlai struck the post with a long-range effort, while Van Dijk later hit the crossbar from a corner as Liverpool pushed for a winner.
Arne Slot introduced Alexander Isak from the bench, but the substitution of Ngumoha - one of Liverpool's brighter performers - was met with audible frustration from sections of the Anfield crowd.
Chelsea, meanwhile, managed to hold firm in the closing stages, with Filip Jorgensen called into action to deny Szoboszlai, while Liverpool were left frustrated by their inability to convert pressure into a decisive goal.
The result means Liverpool remain in control of their top-four hopes but must wait to secure Champions League qualification, while Chelsea avoid setting a new unwanted losing streak and take a rare point from a difficult away fixture.