Back in April their win on Merseyside signalled the beginning of the end of the Reds' title dream and while this 2-1 victory was not as heartbreaking or damaging for the home side it was just as frustrating.
Midfielder Emre Can's first goal since his summer move from Bayer Leverkusen had given Liverpool an early lead but it has not been a situation they have been comfortable with all season.
Gary Cahill's scrambled 13th-minute effort via the goal decision system ensured Chelsea manager Jose Mourinho could have no complaints and he would have been positively overjoyed with Costa's clinical first-time effort midway through the second half.
While Mourinho is sitting pretty at the top of the table his former protege Brendan Rodgers finds himself under more scrutiny having placed particular emphasis on this game by making seven changes for the midweek Champions League defeat at Real Madrid.
The likes of Steven Gerrard, Raheem Sterling and - somewhat surprisingly considering Kolo Toure's performance at the Bernabeu - centre-back Dejan Lovren were recalled with Can the only alteration to the team which lost at Newcastle last weekend.
Having put in a shift on Tuesday the Germany Under-21 international was tasked with adding some solidity to the midfield but what he also did was put himself about.
Only 30 seconds in, his long-range shot was deflected wide by John Terry but the Chelsea captain's centre-back partner Cahill was not as fortunate as in the eighth minute when Can let fly from distance again and his shot ricocheted off the England defender to leave goalkeeper Thibaut Courtois stranded.
While a first goal in three matches was celebrated with unbridled joy at Anfield most in the crowd understood one would not be enough considering their defensive problems and Chelsea's considerable attacking prowess led by Costa.
Gerrard's slip in this fixture last April to present Demba Ba with the opening goal was the beginning of the end for the club's title hopes but when the Liverpool captain was caught in possession by Costa this time he escaped by conceding a corner, which he himself dived to head clear.
Unsurprisingly a set-piece was again to prove Liverpool's undoing as Chelsea's aerial supremacy saw Costa flick Fabregas' corner on to Terry who was denied by a brilliant save by Simon Mignolet only for the Belgium goalkeeper to carry the ball over the line from Cahill's follow-up.
Technology intervened and a goal was awarded.
Sterling and Philippe Coutinho both looked like troubling Chelsea's defence with driving runs but the visitors soon took control with a spell of pressure which instilled almost panic among Liverpool's ranks but failed to bring a second goal
Costa, whose running battle with Martin Skrtel was a continuation of their spat in last month's Croatia versus Spain international, hooked an overhead kick into the Anfield Road end early in the second half with Sterling forcing Courtois into a low save after cutting in from the left.
A succession of niggly fouls and personal confrontations saw the match descend into a spell of scrappiness but that was broken by a turn of pace from Cesar Azpilicueta, back in the side after a three-match ban, and a touch of quality from Costa in the 66th minute.
As the Chelsea left-back burst past Coutinho on the touchline there was a suspicion the ball could have gone out of play but the Spain international continued to drill in a cross which Mignolet could touch to Costa who lashed the ball into the net first time.
Liverpool responded with Jordan Henderson's well-timed volley taking a double deflection off Azpilicueta and Nemanja Matic while Gerrard felt he should have had a penalty when his left-footed shot hit the arm of the ubiquitous Cahill late on.
Even the introduction of Rickie Lambert for Mario Balotelli, who had a first-half effort ruled out for offside but offered little threat otherwise, failed to produce an equaliser despite a slightly more direct approach.
Source : PA