On-loan Hearts marksman Bednar took his goal tally to 12 in 14 games as his 60th minute strike earned Albion their first win in three league games.
But Tony Mowbray's side had to do it the hard way against battling Burnley, who had gone into the game on a high having won their previous three matches.
Former Albion midfielder James O'Connor stunned his old club when he fired Burnley into a third minute lead with his first goal of the season.
But it was an advantage that Owen Coyle's side failed to hold on to with Bostjan Cesar's first goal since arriving on loan from Marseilles last summer dragging Albion level in the 26th minute.
Albion's margin of victory should have been more comfortable as Robert Koren's 68th minute penalty was saved by another Albion old boy Brian Jensen.
Albion again showed what a problem they have defending from set pieces when O'Connor caught them cold.
Robbie Blake's free-kick was nodded down by Stanislav Varga and O'Connor had the simple task of stabbing the ball home from three yards.
Chris McCann was inches away from doubling Burnley's lead two minutes later with a low 20-yard drive before Albion finally came to life.
James Morrison's shot was deflected wide and then the former Middlesbrough midfielder saw his header crash off the bar.
Albion's pressure finally paid off when Jonathan Greening whipped in a corner and the unmarked Cesar made no mistake with a close range volley.
Zoltan Gera's acrobatic overhead kick then whistled narrowly wide in the 32nd minute as Albion ended the half more positively than they had begun it.
Albion took that momentum into the second half and were rewarded on the hour when Paul Robinson's left-wing cross was headed home at the far post by Bednar.
Burnley's challenge should then have been extinguished once and for all when Koren was sent sprawling by a Jon Harley challenge.
But Koren's spot-kick was weak and Jensen dived low to his left to beat away the penalty.