Eden Hazard’s penalty on 57 minutes appeared set to lift Chelsea back into third spot after Liverpool’s half-time substitute Daniel Sturridge had cancelled out Oscar’s first-half header.
However, midway through the second half, Suarez appeared to bite Branislav Ivanovic before the Uruguayan striker headed home an equaliser in the seventh minute of added time.
Liverpool created the first chance on five minutes when Suarez’s clever pass slipped in Glen Johnson, but the former Chelsea full-back could only poke his shot wide.
Jordan Henderson then volleyed over from 25 yards before Chelsea tested Pepe Reina for the first time on 20 minutes when the Liverpool keeper punched away Ramires’ dipping effort.
Chelsea went ahead on 26 minutes when Oscar took advantage of slack defending to head Juan Mata’s corner inside the post and the lead was almost doubled three minutes later.
Henderson was booked for bringing down Ramires and from the resulting free-kick David Luiz, who scored a spectacular goal at Fulham in midweek, saw his long-range drive fumbled by Reina, who recovered in time to prevent the ball rolling over the line.
Suarez saw a shot beaten away by Petr Cech before half-time and Liverpool then made a blistering start to the second half following the introduction of Sturridge for Philippe Coutinho.
Within 15 seconds of the restart, Sturridge had slipped in Steven Gerrard, whose shot was kept out by Cech’s legs, and a minute later the former Chelsea forward rattled the post after a smart turn and shot.
Sturridge then drew Liverpool level on 52 minutes when he was presented with a simple finish at the far post from Suarez’s superb cross after Stewart Downing had neatly flicked the ball back to the Uruguayan.
However, Suarez then gifted Chelsea the lead again on 57 minutes when he was penalised for handball from a corner and Hazard nonchalantly sent Reina the wrong way from the spot.
Midway through the second half and a tangle between Suarez and Ivanovic resulted in the Liverpool striker appearing to bite the Chelsea defender on the arm in an incident which is likely to be reviewed by the Football Association.
Substitute Jonjo Shelvey side-footed a great opportunity to equalise for Liverpool wide with five minutes to go and Sturridge was fortunate to escape punishment for a challenge on Ryan Bertrand.
With Chelsea looking set to hold on for a much-needed win, Sturridge crossed from the right deep into stoppage-time and Suarez’s near-post header beat Cech.