The hosts were given hope of a huge upset when Scott Arfield put Burnley in front in the 14th minute, but it was all Chelsea after that. Costa showed why Chelsea spent 32 million pounds ($53 million) to bring him in from Atletico Madrid when he slotted in the equalizer just three minutes later, and Fabregas set up Andre Schuerrle with a sublime pass in the 21st as the German World Cup winner made it 2-1.
Fabregas then sent in a corner in the 34th that Branislav Ivanovic volleyed into the net after some slack defending.
Costa had earlier had a penalty appeal waved away and was booked for diving in the 31st, despite replays showing that his right foot was clipped by onrushing Burnley goalkeeper Tom Heaton.
Costa and Fabregas were the two major offseason recruits for Jose Mourinho's team, and they showed right away why their arrivals have elevated Chelsea into being the bookmakers' favorite for the title.
Fabregas, especially, was involved in nearly every Chelsea attack on his return to the Premier League - and may have given Arsenal manager Arsene Wenger second thoughts about his decision not to re-sign his former captain from Barcelona this summer.
''He really was the maestro, he controlled the tempo of the game,'' Mourinho said. ''He was the man that dictated the speed, the intensity and the direction of the game.''
Fabregas' standout moment came in the buildup to Chelsea's second goal when he finished off a 25-pass team move with an inch-perfect ball into the area for the onrushing Schuerrle to slot inside the far post.
''So far so good,'' Fabregas said. ''I think we have a very talented young team.''
While the game was a bright start to the new season for Chelsea, it also may have signaled the end of an era as Mourinho picked Thibaut Courtois as his starting goalkeeper ahead of Petr Cech, who has been the Blues' No. 1 for the last decade.
Courtois was brought back after an impressive loan spell at Atletico Madrid and looked assured on his debut, although he had no chance of stopping Arfield's early strike.