In a fiery atmosphere, the home side took a surprise lead in the 14th minute with a superb drive from Scott Arfield after a fine pull back from Matt Taylor.
But Chelsea quickly got back on level terms, three minutes later, with Brazilian-born Spain international Costa, a close season signing from Atletico Madrid, scoring on his English Premiership debut - firing in after a Branislav Ivanovic cross-shot had come out off the post.
Within four minutes, Jose Mourinho's side had the lead - Fabregas providing a beautiful first time pass to German World Cup winner Andre Schurrle who confidently drove home.
Burnley looked shell-shocked and their manager Sean Dyche will be disappointed with the defending for Chelsea's third as Ivanovic was unchallenged as he slotted in a Fabregas corner in the 34th minute.