Sebastian Larsson's free-kick in the dying seconds of injury time was no more than a consolation effort for the Blues.
Burnley captain Steven Caldwell started his first game of the season, which allowed Bikey to play in a central midfield position.
Blues boss Alex McLeish maintained his normal formation, which meant a lone striker in the form of Garry O'Connor.
The first half saw little goalmouth action with Blues goalkeeper Joe Hart only having to make one save from Burnley striker David Nugent.
At the other end Burnley goalkeeper Brian Jensen made his only save from O'Connor. The best opportunity in the opening 45 minutes fell to veteran midfielder Lee Bowyer, who should have done better from close range following a cross from Stephen Carr.
McLeish made a double substitution at half-time introducing Larsson and Liam Ridgewell, but it was the home side who scored with a swift counter-attack led by right-back Tyrone Mears and finished by Fletcher - for his first goal in the Premier League since joining in the summer from Hibernian.
Fletcher should gave scored again shortly afterwards but after being sent clear by Robbie Blake, he saw his shot strike a post.
The home side increased their lead when Bikey played a lovely one-two with on-loan striker Nugent and finished in some style.
Goalkeeper Hart kept his side in the game with saves from Fletcher and Mears, but all the Blues had to show for their efforts was an injury time free-kick expertly executed by Larsson.
The scoreline may suggest a narrow home victory, but the reality was that once the Clarets had gone in front, there was never any doubt that they would maintain their impressive record at Turf Moor.