The Brazil-born Spain striker claimed his 10th goal of the campaign on 67 minutes by lashing home in lethal fashion after goalkeeper Simon Mignolet had turned a cross from Cesar Azpilicueta into his path.
The goal completed Chelsea’s fightback after Gary Cahill had cancelled out a ninth-minute goal by Emre Can and strengthened Jose Mourinho’s side’s grip on first place.
Liverpool had been hoping to erase memories of the loss in last season’s corresponding game that effectively cost them the league title and Can’s first goal for the club gave them an ideal start.
Steven Gerrard, whose slip in that April fixture allowed Demba Ba to put Chelsea ahead, found Raheem Sterling on the left, and he in turn picked out Can.
The former Bayer Leverkusen midfielder was allowed to advance before producing a 20-yard shot that took a wicked deflection off Cahill and wrong-footed Thibaut Courtois on its way into the net.
It was just what Rodgers’s side needed to calm early nerves and silence the pre-match taunts of Mourinho, who had taken obvious pleasure in referring to last season’s fixture before the renewal of hostilities.
But on this occasion Liverpool could only hold on to the lead for five minutes.
The league leaders levelled when Cesc Fabregas’s corner was turned goalwards by Costa and John Terry’s header forced an instant reflex save from Mignolet.
With the Liverpool defence static, Cahill pounced and despite Mignolet making a strong effort to keep the ball out, goal-line technology, relayed to referee Anthony Taylor’s wristwatch, proved it had crossed the line.
– Late penalty claim –
Liverpool were not immediately affected by the equaliser.
Mario Balotelli had a goal ruled out for a clear offside, Sterling saw a shot hit Cahill on the shoulder, and Philippe Coutinho forced Courtois to save low to his right after a strong run and shot.
But it was not long before Mourinho’s team, unbeaten in the league since last March, were clicking through the gears.
Costa and Martin Skrtel were waging a physical battle in the Liverpool half and they clashed on their way down the tunnel at half-time, obliging their team-mates to intervene to prevent the matter escalating.
Eden Hazard had at leat four shots blocked by defenders as Liverpool struggled to get out of the last third, while Costa found himself in shooting positions twice without seriously testing Mignolet.
It was a far cry from last season’s equivalent fixture, when Rodgers had accused Mourinho of having “parked two buses” in front of the Chelsea goal in an effort to frustrate Liverpool.
The pattern continued after the restart, with Costa’s spectacular bicycle kick clearing the bar as he connected with Dejan Lovren’s defensive header.
Sterling replied with a stinging shot that Courtois saved impressively, but Chelsea were soon back on the front foot. Hazard beat Glen Johnson too easily before advancing and seeing his shot blocked yet again by a red shirt.
The right flank was a problem area for Liverpool, and it was from there that the second Chelsea goal came.
Thereafter, there were few genuine moments of hope for the home side as they sought an equaliser, although Gerrard’s late shot drew another strong, but unsuccessful, penalty claim for a handball by Cahill.