The Spain international striker netted a second-half winner for the Blues after Gary Cahill had cancelled out Emre Can's opener at an emotionally-charged Anfield
The Blues had to come from a goal down on Merseyside, with Emre Can’s deflected strike giving the home side the lead, and Costa pounced with 20 minutes to go following Gary Cahill’s equaliser.
|VIEW FROM ANFIELD
|By Tom Maston
Brendan Rodgers will no doubt point to refereeing decisions that did not go his team's way, but after a bright opening ten minutes Liverpool never got going and Chelsea more than deserved their three points.
In Oscar and Eden Hazard the Blues have two of the hardest working attacking midfielders in the division, and after a shaky start Gary Cahill and John Terry showed all their class in defence.
The Reds desperately need Daniel Sturridge back after another poor showing from Mario Balotelli, and questions will now be asked as to whether the team picked to play Real Madrid was worth it.
But they again shot themselves in the foot, with slack defending allowing Cahill and Costa to overturn Can's first Liverpool goal and move Chelsea a daunting 15 points clear of the Merseysiders.
Much had been made of Rodgers' team selection away to Real Madrid in midweek, with the likes of Gerrard, Mario Balotelli and Jordan Henderson recalled after being omitted from the starting line-up for the Champions League clash.
Yet it was one of the supposed second string to have impressed in the Spanish capital, Can, who put the hosts ahead early on with an effort that deflected in off Cahill.
The England man - perhaps fortunate to twice go unpunished for handball in the Chelsea area - atoned just five minutes later, though, bundling home despite Simon Mignolet's best efforts.
Jose Mourinho's approach was much more attacking than the April smash-and-grab - a reward coming after 67 minutes as a string of Liverpool lapses at the back were punished by Costa's clinical finish.
Still undefeated, Chelsea have now taken five points from a possible nine away to Manchester City, Manchester United and Liverpool to firmly underline their credentials as England's best this campaign.
Can was one of just three outfield players retained from the Madrid fixture and gave Liverpool a ninth-minute lead thanks to a sizeable nick off Cahill.
But Cahill quickly had Chelsea level, reacting quickest to prod towards goal after Mignolet had sharply denied John Terry from a corner, with technology confirming the Belgian had carried the ball over the line.
An already highly-charged atmosphere was ramped up as claims for a Cahill handball in the penalty area fell on deaf ears before Costa introduced himself to the contest by shoving Martin Skrtel towards an onrushing Mignolet, earning a warning from referee Anthony Taylor.
A semblance of composure returned to the game but Chelsea looked increasingly dangerous on the break - especially behind the adventurous Alberto Moreno at left-back for Liverpool - as they built pressure prior to the interval.
The second period began at much more sedate tempo, Costa's wayward overhead effort and Thibaut Courtois' save from Raheem Sterling representing rare flashpoints.
Yet the hosts were perhaps the architects of their own downfall as Chelsea moved ahead.
Substitute Willian found Cesar Azpilicueta on the left flank, with Glen Johnson out of position, and Philippe Coutinho did little to halt the Spaniard's progress to the byline.
Azpilicueta's cut-back was palmed into a dangerous area by Mignolet, allowing Costa to power home his tenth league goal of the season.
With Fabio Borini and Rickie Lambert thrown on as Liverpool sought a dramatic equaliser, a blatant Cahill handball went unnoticed in the dying embers, leaving Mourinho to calmly acknowledge victory in stark contrast to his tub-thumping celebrations from last term.