Paul Ince's 15th-minute strike was enough to give battling Wolves a hard-fought victory at Turf Moor.
Chris McCann came close early on for the Clarets when he rifled over, then Carl Cort should have given Wolves the lead but mis-timed his header.
Cort then made amends when he set up Ince, whose crisp shot was far too powerful for keeper Brian Jensen.
Burnley searched for an equaliser after the break, with Michael Ricketts going close, but the visitors held on.
Burnley manager Steve Cotterill:
"It was disappointing in the end because we felt as if we'd done to get something out of the game.
"We could have had a couple of penalties , so we're disappointed.
"When we play the bigger teams the referees don't want to give anything against them and that's what happens. All of the lads were desperate to win."
Wolves boss Glenn Hoddle:
"It was a great move that led to the goal and the strike was vintage Incey.
"Paul pulled his hamstring scoring but it shows his character that he carried on and played the full 90 minutes.
"This win will give us confidence. We will go back to Molineux feeling buoyant. This has given us a lift - we still believe we can do it."
Burnley: Jensen, Elliott, Duff, McGreal, Harley, Spicer, James O'Connor, Hyde, McCann (Branch 63), Noel-Williams, Ricketts.
Subs Not Used: Karbassiyoon, Garreth O'Connor, Courtney, Pugh.
Booked: Ricketts, McGreal.
Wolves: Postma, Ross, Lescott, Craddock Naylor, Davies, Ince, Rosa (Anderton 75), Cort, Frankowski (Aliadiere 80), Miller.
Subs Not Used: Oakes, Kennedy, Ganea.
Booked: Rosa, Ross.
Goal: Ince 15.
Ref: C Oliver (Northumberland).