Bednar's third goal of the season - but his first from open play - in the 61st minute sentenced Fulham to a fourth straight defeat.
It also extended their poor record on the road as they have lost all three of their Premier League games away from Craven Cottage.
Had they been more clinical in front of goal in the first half then things could have been a lot different for Roy Hodgson's side.
But Albion proved more of a threat after the break and their margin of victory would have been greater but for the Fulham goalkeeper Mark Schwarzer.
He pulled off impressive saves from Bednar and Robert Koren, but by then the damage had been done.
Bednar's goal made amends for an early miss as he wasted a golden chance to hand Albion the initiative after just eight minutes.
Skipper Jonathan Greening won a midfield battle and fed a pinpoint pass into the path of Bednar, but he took too long over his shot and his tame left foot drive was comfortably collected by Schwarzer.
While Albion may have enjoyed the greater possession, it was Fulham who carried the threat and they should really have turned that into a healthy advantage.
But their finishing left something to be desired with the lively Clint Dempsey being denied by Scott Carson in the 13th minute, before Jimmy Bullard fired a 20-yard free-kick just over the bar two minutes later.
Albion struggled to keep Dempsey in check and he was again close to breaking the deadlock from long range in the 23rd minute, before turning provider for Albion old boy Zoltan Gera.
Gera, on his first return to The Hawthorns since leaving in the summer, was teed up just inside the area by Dempsey but saw his low drive flash just wide with Carson well beaten.
Fulham were to rue their failings in front of goal as Albion added menace to their play in the second period.
Ryan Donk crashed his close-range header against the post from a 47th minute Jonathan Greening corner and it was a warning Fulham should have heeded.
When James Morrison's header was cleared off the line by Danny Murphy, Greening delivered another corner that Fulham again failed to deal with.
Once again the ball found its way to Donk and this time he rolled the ball across a crowded goalmouth and into the path of Bednar, who made no mistake from six yards.
Fulham, who saw Murphy fire just wide as they went in search of an equaliser, were fortunate not to fall further behind in the 69th minute when only a stunning save by Schwarzer kept them in contention.
Koren's right-wing cross was deflected goal wards by Bednar from just six yards only for Schwarzer to instinctively deflect the ball away.
Schwarzer then denied Koren low at his near post before Bednar saw his 86th minute drive from 30 yards crash off the bar.