The Brazil-born Spaniard - who was a fitness doubt for the game with a minor muscular injury - continued his prolific scoring form as goals reigned supreme at Goodison Park
Costa and Branislav Ivanovic struck inside three minutes to put Jose Mourinho's men in a commanding position but a flying Kevin Mirallas header brought the hosts back into contention before the interval.
A Seamus Coleman own goal on 67 minutes put Chelsea back in control before Nemanja Matic, Samuel Eto'o and Ramires struck within the space of four incredible minutes, but Costa netted again at the death as the Blues held on to preserve their 100 per cent start to the season.
|VIEW FROM GOODISON PARK
|By Peter Staunton
Where to begin with that? It's been a long time since a game rocked from start to finish like that in the Premier League.
And Everton may have wished his name had not been on the team sheet when Costa collected Cesc Fabregas' pass and sprung the offside trap before sliding the ball between the legs of a woefully exposed Howard after just 35 seconds.
Things went from bad to worse for Roberto Martinez's side when Ramires forced the ball through a disorganised home defence to pick out Ivanovic, who in turn found the back of the net - although he looked to have strayed beyond the last defender.
Amazingly, the early drama continued as Howard raced out to collect Eden Hazard's searching long ball and handled outside his area, but the American goalkeeper escaped punishment.
The hosts thought they had pulled a goal back in the 15th minute after former Chelsea striker Romelu Lukaku's header struck the crossbar and was turned in by Distin, but the defender was offside.
Distin was then relieved to see a Fabregas strike flash just wide after the Frenchman had got a toe-end on the ball in the 29th minute.
Everton responded, and Mirallas brought them back into the game with an excellent header from Coleman's cross in first-half stoppage time.
Phil Jagielka's careless pass inside his own half almost sent Everton back to square one within seven minutes of the restart, but Costa was thwarted by Howard after racing through on goal.
Chelsea's third did arrive in the 68th minute when Hazard's attempted cutback was turned into his own net by Coleman, only for Naismith to hit back inside 60 seconds with a cool finish after getting in behind a static visiting defence.
Matic's measured strike from distance kept Chelsea on top, though their lead was narrowed when Eto'o seized his moment to keep the pressure on with a close-range finish.
Remarkably, that was not the end of the scoring, as Ramires found the bottom corner with 13 minutes left and then substitute Muhamed Besic's poor first touch handed Costa his second, completing a memorable Chelsea victory.