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Guus Hiddink's side began the day in 16th place, three points above the relegation zone after a woeful opening half of the season, but comprehensively outplayed seventh-placed Palace in torrential rain at Selhurst Park.
Dutchman Hiddink is unbeaten since taking over as manager on an interim basis following the sacking of Jose Mourinho last month, although the win at Palace was his first following draws with Watford and Manchester United in the Christmas programme.
Chelsea's third victory in their last 11 league matches moved them up two spots on 23 points, 13 points behind fourth-placed London rivals Tottenham Hotspur.
Former Spurs winger Aaron Lennon gave Everton the lead but Deli Alli equalised on the stroke of halftime as the Spurs stretched their unbeaten away run to nine games.
Tottenham have 36 points, six behind leaders Arsenal, who moved top with a 1-0 home win over Newcastle Unitedat the weekend.
While a top-four finish still appears unlikely for Chelsea, captain John Terry said the club was now moving in the right direction having lost nine of their opening 15 games.
"This was the Chelsea of last year, the football was first class and we kept the ball in tough conditions," captain Terry told Sky Sports.
"A near year, a new start, we just have to keep winning games now. We are where we are, we knew as a group we had to get better and we have done so slowly and surely and today hopefully is not the last of it."
Palace, who could have gone fifth with a win, were without two of their most influential players in suspended midfielder Yohan Cabaye and injured winger Yannick Bolasie but more than matched the visitors for 25 minutes.
Frazier Campbell had a half chance but sliced wide and shortly afterwards Chelsea struck.
Cesc Fabregas, booed by Chelsea's fans in the wake of Mourinho's dismissal, split the Palace defence with a measured pass and Costa, another player who has disappointed this season, advanced on goal before unselfishly squaring the ball for Oscar to stroke home after 29 minutes.
Palace keeper Wayne Hennessey saved well from Cesar Azpilicueta soon afterwards as Chelsea began to pop the ball around confidently on the sodden turf.
Wilfried Zaha wasted Palace's only opening of the second half when he shot tamely at goalkeeper Thibaut Courtois.
Chelsea then extended their lead in spectacular fashion on the hour, Willian's ferocious rising drive from just outside the penalty area giving Hennessey no chance.
Six minutes later Costa capped a hard-working performance with a close-range third goal after Willian's shot was saved.
Chelsea's only concern was an injury to their Belgium forward Eden Hazard who limped off midway through the first half after signalling to the bench that he could not continue.