Eden Hazard scored what had looked to be the 58th minute winner from the penalty spot after Daniel Sturridge had cancelled out Oscar’s first-half header.
But Suarez headed in Sturridge's cross just as Chelsea appeared to have secured three points to keep the club on course for a top-four finish.
On an afternoon when Anfield paid tribute to former manager Rafael Benitez, back at the stadium for the first time since he was sacked in 2010, and tireless Hillsborough campaigner Anne Williams, the occasion will also be remembered for the less savory side of Suarez.
The Liverpool forward appeared to bite Branislav Ivanovic on the arm midway through the second period after they had jostled for the ball in the Chelsea penalty box.
The Uruguayan, who was banned during his Ajax days for biting, is likely to find himself in hot water when the footage of the incident is reviewed by the Football Association.Benitez has had to juggle his resources to cope with Chelsea’s fixture overload but he made a comparatively meager three changes from the team that won 3-0 at Fulham on Wednesday. Cesar Azpilicueta, John Obi Mikel and Oscar came in for John Terry, Frank Lampard and Victor Moses, who all dropped to the bench.
Liverpool made only a single alteration to the team that drew 0-0 at Reading last weekend, with Stewart Downing replacing the benched Daniel Sturridge.
The air was heavy with emotion and sentiment at Anfield. Before kickoff there was a minute’s applause in memory of Hillsborough campaigner Anne Williams, who died during the week, and the Boston Marathon. In a terrific tribute, ‘There’s only one Anne Williams’ reverberated around Anfield.
Meanwhile, Benitez was given the warmest possible welcome by the Kop, where the fans chanted his name throughout the first half.
The former Liverpool manager watched his old side take an early grip of the match without looking particularly threatening, but it was the team he manages only on an interim basis which took the lead after 25 minutes through an unlikely source.
Oscar deserves credit for the deftness of his header direct from Juan Mata’s corner but there will be question marks over both the failure of the defenders to track his straightforward run to the front post and the goalkeeping of Pepe Reina, who was beaten inside his near post.
Moments later, Reina nearly made one of the howlers of the season. A speculative David Luiz shot was pinged straight at him but the Spaniard contrived to make an almighty mess of it, almost spilling it behind the line. The sighs of relief around Anfield were audible when he clawed it away at the last second.
Suarez always looked Liverpool’s most likely avenue for a goal but it was not until the 42nd minute that the Uruguayan got in sight of goal. The forward was slipped in by Downing on the right of the area but his fiercely-struck shot was too close to Petr Cech, who comfortably made his first meaningful save of the afternoon. The same players were in action on the cusp of halftime but, on this occasion, Suarez shot weakly at the keeper from what was a tight angle.
Sturridge, a halftime replacement for the disappointing Philippe Coutinho, immediately gave Liverpool the spark the team had been missing.
The former Chelsea man’s first two contributions were exquisite. First, he nearly created the equalizer with a brilliant run and pass from the right only for Cech to divert away Gerrard’s first-time shot with his leg. Then, he cracked a superb 30-yard shot against the post after pirouetting away from the attentions of the boys in blue
It was third time lucky for Sturridge, who put the finishing touch to a sublime move after 52 minutes following the most delightful of dinked crosses from Suarez.
With Liverpool playing some sparkling stuff, the atmosphere was vibrant. But the home side could only stay level for five minutes after conceding another soft goal.
Replays showed referee Kevin Friend was correct to point to the penalty spot after Suarez handled the ball in his own area. Hazard nonchalantly drove the spot kick into the left corner.
Suarez was at the heart of all Liverpool’s best play but he was once again cast in the role of villain midway through the second period. After wrestling for the ball with Ivanovic in the Chelsea box, Suarez grabbed the defender’s arm, moved his face towards it and appeared to sink his teeth into his opponent’s bicep. It instantly looked a reckless act.
Liverpool continued to press for an equalizer. Jonjo Shelvey appeared to have wasted the club's final chance when he shot wide late on. But Suarez refused to give up and sent the home support wild with his late headed equalizer from Sturridge's cross.