Former Arsenal midfielder Fabregas, brought in from Barcelona during the close season, was influential throughout the game, having a hand in all three Chelsea goals and striking up an impressive midfield pairing with Eden Hazard.
Chelsea recovered from conceding a 14th minute goal, quickly turning on the style in an impressive half hour spell to go in 3-1 up at the break.
Jose Mourinho's side were content to see out the game in the second half as Burnley, tipped by many for relegation, put in a creditable professional display after the Londoners had threatened to run away with the game.
"I think we had a good reaction and the first part of that is that emotionally we were not affected by the goal we conceded," said Mourinho.
"We kept playing according to the way we want to play, the players were calm and never lost their composure and after that we had a fantastic 20-25 minutes with three goals, the second one a fantastic goal, and in the second half we had control of the game," he said.
"I was a bit afraid because I know that my players are not still in the top physical form, ready to give the same kind of performance for 90 minutes so we lost a little bit of attacking appetite but the control of the game, of the possession was always there and I think we clearly deserved this three points against a side that is going to give a fight to everyone who comes here," added the former Real Madrid coach,
In a fiery atmosphere, the home side took a surprise lead in the with a superb drive from Scott Arfield after a fine run and pull back from Matt Taylor.
But any hopes the capacity home crowd at Turf Moor had of witnessing an upset were quickly dashed as Chelsea showed why so many bookmakers have them as favourites to win the title this season.
Three minutes after Arfield's strike, Brazilian-born Spain international Costa, a close season signing from Atletico Madrid, brought the Londoners level on his Premier League debut - firing in after a Branislav Ivanovic cross-shot, following a Fabregas back-heel, had come out off the post.
Four minutes later, Chelsea 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 slack defending for Chelsea's third as an unchallenged Ivanovic slotted in a Fabregas corner in the 34th minute.
Mourinho had raised some eyebrows by opting for Thibaut Courtois over Petr Cech in goal and the Belgian showed his quality shortly after the break to deny Arfield a second from a curling shot which he tipped wide.
Lancashire club Burnley, whose entire squad cost just a fraction of Costa's reported 32 million pound transfer fee, will have taken some heart though from the way they applied themselves after the break, even if they rarely threatened to alter the scoreline.
Chelsea brought on Didier Drogba for his first official appearance of his second spell with the club as a late substitute to the delight of their travelling supporters.
Dyche said his team could only learn from playing against a top quality opponent so early in the campaign.
"There has to be a reality to what we are hoping to achieve this season. I think they are top of the market or certainly up there.
"Overall I felt we kept going which was important, that mentality will be important over the course of the season, the will and demand was there. It is fair to say that they have so much quality in all areas and they can hurt you in any way," he said.