The Spain striker, who opened his account with a poacher's goal on his debut in the Blues' 3-1 opening victory at Burnley on Monday, pounced again from inside the box to put Jose Mourinho's side ahead against the promoted visitors just after the hour.
Eden Hazard added a second goal 13 minutes from time to put the result beyond doubt, with the biggest cheer of the day reserved for the late introduction of Didier Drogba in place of Costa.
Mourinho had made another big selection call in leaving striker Fernando Torres out of the match-day squad, Torres has been linked with a move to Italy, but Mourinho says he won't off-load the Spaniard.
"The reason is I decided to keep the same team, to have a bench with a complete balance. Sometimes I can have two strikers on the bench, sometimes I can't," he said.
"I want three strikers in the team. Sometimes I play with two, one on the bench, sometimes someone will be injured or suspended. I cannot do a season without three strikers. I want him [to stay].
"He will play and score goals, he is a fantastic guy, a fantastic professional, so there is no problem."
Mourinho added that he had been impressed by the speed of Brazilian-born Costa's transition to English football after his summer £32 million transfer from Athletico Madrid.
"I got the feeling in pre-season that his adaptation would not be a big deal, he is an easy guy," said Mourinho.
Managerless Crystal Palace went down 3-1 at home to West Ham.
Mauro Zarate, Stewart Downing and Carlton Cole were the men on target for Sam Allardyce's side, who notched their first victory of the season following their last-gasp home defeat to Tottenham.
Marouane Chamakh had halved the deficit for the Eagles just after the break, but it proved to be in vain.
"One result doesn't determine if you're the right manager or not," said Palace caretaker manager Keith Millen.
"I know from our conversations what the chairman Steve Parish is looking for, and if he feels I'm the right one to do it, then I will talk to him again. We have spoken all week really, because we know it's coming to the deadline now and we haven't appointed anyone yet."
However, ex-Celtic boss Neil Lennon remains the bookies' favourite at 7/4.
Nathan Dyer's goal earned Swansea all three points against Burnley at the Liberty Stadium. The winger latched on to a through-ball from Gylfi Sigurdsson and coolly slotted home midway through the first half to continue the Swans' 100 per cent start following last weekend's success at Manchester United.
Southampton and West Brom remained in search of their first wins after battling out a goalless draw at St Mary's.
In the lunchtime kick-off, 10-man Newcastle fought out a 0-0 draw with Aston Villa. The game at Villa Park saw Magpies defender Mike Williamson sent off having collected two yellow cards in quick succession.