The Brazil-born Spain striker cost Chelsea manager Jose Mourinho £32 million ($53.1 million, 40.4 million euros) when he joined from Atletico Madrid in the close season and he provided a further illustration of his worth at Goodison Park.
The 25-year-old started and finished the scoring on an extraordinary evening on Merseyside, taking his tally to four goals in three games since his arrival at the club.
Everton's own big-money striker Romelu Lukaku, a £28 million acquisition from Chelsea, lined up against his old club, with another Chelsea old boy, Samuel Eto'o, starting the game on the bench.
Mourinho included Costa in his starting XI after he passed a late fitness test, having not trained in the latter half of the week due to a hamstring complaint.
And the striker showed no ill-effects to give Chelsea the lead after 36 seconds when he latched onto a through-ball from Cesc Fabregas.
Costa let fly with a low angled shot that Everton goalkeeper Tim Howard got a foot to but was unable to prevent going into the net.
Chelsea added a second goal in the third minute when Everton's appeals for offside were ignored by the officials.
Neat interplay between Costa and Brazil midfielder Ramires allowed the latter to slip the ball into Branislav Ivanovic, who had ghosted in behind the Everton defence and clinically dispatched a low shot past Howard.
American goalkeeper Howard was fortunate not to be punished by referee Jon Moss shortly afterwards when he handled outside the area as Eden Hazard chased down a long ball from Fabregas.
Everton improved as the first half wore on. Lukaku rattled the crossbar with a header from a corner and Sylvain Distin bundled the ball into the net, but the assistant referee had already correctly raised his flag for offside.
Everton pulled a goal back on the stroke of half-time when Seamus Coleman was released down the right flank and crossed for Kevin Mirallas, who got in front of Gary Cahill to glance a superb header into the top-left corner.
The goal ensured Chelsea were unable to rest on their laurels as a thrilling second half ensued.
Chelsea added a third in the 67th minute when Hazard cut in from the left and saw his low cross-shot inadvertently diverted past Howard at the near post by the sliding Coleman for an own goal.
But Everton refused to lie down and two minutes later reduced the deficit to one goal when Aiden McGeady jinked his way forward and threaded the ball to Steven Naismith, who stabbed a shot past Thibaut Courtois with his right foot.
Chelsea regained the initiative shortly afterwards with their fourth goal when Nemanja Matic arrowed a drive from the edge of the area into the bottom-right corner of the net via a deflection off Distin.
But Everton's never-say-die attitude saw them pull it back again with their third when a Leighton Baines free-kick from wide on the left picked out substitute Eto'o and he directed a clever header past Courtois.
Chelsea bagged their fifth goal of an engrossing contest when Ramires cut in from the left, played a neat one-two with Matic, and poked the ball past Howard.
And Costa made doubly sure that there would be no late drama when he seized onto a clever back-heel from John Mikel Obi, drew Howard from his line, and drove the ball into the net with his left foot.
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