Wolves will certainly feel that this was an opportunity missed as they controlled the game for long spells - most notably in a frantic opening spell.
Albion only really came into the game in the second half and will be relieved to be heading into Wednesday's second leg with an advantage.
Wolves, who suffered a major pre-match setback when goalkeeper Matt Murray was ruled out with a broken shoulder, leaving rookie Wayne Hennessey to step in for his debut, made all the early running but could not make their pressure pay.
Michael Kightly was denied by goalkeeper Dean Kiely, Seyi Olofinjana had his header cleared off the line by Robert Koren and Neill Collins stabbed the ball wide.
Albion made Wolves pay for their profligacy by taking the lead in the 25th minute following a mistake by Gary Breen.
Breen's attempted clearance after a centre from Koren flew straight to Kevin Phillips who made no mistake with a curling 20-yard drive.
Despite the shock of falling behind, Wolves remained dominant and, after Kightly had again been denied by Kiely and Jay Bothroyd had headed over the bar, their pressure was rewarded on the stroke of half-time.
Albion beat away Bothroyd's free-kick and Michael McIndoe sliced a misplaced cross into the penalty area.
It somewhat fortuitously flew to the unmarked Andy Keogh whose cross was powerfully headed home at the near post by Jody Craddock.
Wolves then looked to have put themselves on course to take control of the tie when Olofinjana gave them the lead six minutes into the second half.
Once again a Bothroyd shot caused chaos. Jackie McNamara floated the loose ball back into the area and a knock-down by Collins was forced home from close-range by Olofinjana.
But any hopes that Wolves had of extinguishing the Albion challenge disappeared three minutes later.
Phillips took full advantage of being unmarked and headed home Darren Carter's 54th minute corner from close-range.
Kamara, who had barely threatened before, then gave Albion a crucial edge for the second leg by outpacing Craddock and drilling his shot past Hennessey.
Kiely then ensured Albion kept their advantage with a smart injury-time save to deny Kightly for the third time.