The Clarets forced a couple of early corners, taken by Robbie Blake, the second of which presented a free header for Chris McCann but he failed to keep his effort down and headed over.
The pattern was repeated as right-back Michael Duff then headed Blake's next corner beyond the crossbar.
Albion took the lead with a swift counter-attack from a further Blake corner which resulted in striker Kevin Phillips stepping inside defender Stephen Jordan and curling his shot around the dive of new keeper Gabor Kiraly.
Jordan was then involved at the other end picking out Duff with a cross but the Northern Ireland International couldn't trouble Albion keeper Dean Kiely with his shot.
The Clarets suffered a blow on the half-hour as skipper Steven Caldwell was forced off with an injury and replaced by Kyle Lafferty.
Zoltan Gera was twice involved in quick succession for Albion, seeing his shot deflected for a corner from which his header brought a save from Kiraly.
Burnley equalised just two minutes into the second half as Michael Duff headed a Blake corner beyond Kiely.
Felipe Teixeira went close to restoring the visitors' lead with a long-range effort after good work from athletic left-back Miguel Angelo Tininho.
Duff was again involved in Burnley's next opportunity, heading the ball back across goal where striker Andy Gray headed wide.
Substitute Ade Akinbiyi should have done better with a great chance as he tamely headed over a cross from Jon Harley.
The winner came 11 minutes from time as Akinbiyi's shot was adjudged to have been handled by central defender Neil Clement and Gray converted the resultant spot-kick.
Even then the action wasn't over as Gera came close to an injury-time equaliser but his shot was pushed behind by keeper Kiraly.