The two sides came into the clash having won all three of their Premier League matches so far this season, but it was Chelsea who maintained their perfect record as they came from behind to secure another three points.
Swansea looked the brighter in the early stages and the Welsh club took an unexpected lead courtesy of a John Terry own goal in the 11th minute at Stamford Bridge.
Chelsea began to turn the tide and were level through the in-form Costa, who headed in a Cesc Fabregas corner before the interval.
The Spain international popped up again in the 56th minute to cap a well-worked passing move, before rounding off his hat-trick 11 minutes later - his seventh goal in four matches.
Debutant Loic Remy added a fourth in the 81st minute with a low strike beyond Lukasz Fabianski after coming on as a substitute.
Jonjo Shelvey then grabbed Swansea's second goal but it was too little, too late as Jose Mourinho's side took their tally to 10 goals in their last two top-flight games to go three points clear at the top of the table
Wilfried Bony dropped to the bench after returning late from international duty with Ivory Coast, while Costa led the Chelsea line having missed out on Spain's Euro 2016 qualifying opener against Macedonia with a hamstring injury.
It was Premier League player of the month Costa who got the first sight of goal, but his deflected effort from just outside the penalty area was comfortably collected by goalkeeper Fabianski.
But Garry Monk's men took a shock lead in the 11th minute – Terry diverting the ball into his own net as he attempted to clear a Neil Taylor cross from the left.
The advantage was almost doubled soon after as Bafetimbi Gomis – who replaced Bony – powered forward before firing wide of the right-hand post.
Terry headed over from a free-kick at the right end as he looked to atone for his earlier error, and Chelsea's frustrations became apparent when Costa appeared to lash out at Gylfi Sigurdsson in an off-the-ball incident.
Swansea had to withstand considerable pressure towards the end of the half and Costa eventually found a way past Fabianski just before the interval, by heading home from a Fabregas corner.
Federico Fernandez replaced Jordi Amat at half-time to make his Premier League debut, and had to be alert from the off as Chelsea continued to look dangerous.
But the Swansea defence was breached once again in the 56th minute when Eden Hazard and Fabregas exchanged passes brilliantly before the latter set up Costa for a simple side-footed finish.
And Costa completed his hat-trick 23 minutes from time by losing his marker to get on the end of a Ramires through ball and sneak an effort past Fabianski, before Remy made it four nine minutes from time with a clinical finish into the bottom corner of the net.
Shelvey slid the ball beyond goalkeeper Thibaut Courtois after being sent through on goal in the closing stages, but it proved to be a mere consolation for the visitors.
