The result, which sealed a first double over Wolves since 1957 for Sean Dyche's side, leaves them three points adrift of safety with only one game remaining.
Goals in either half from Danny Ings and Martin Paterson were enough to see them home, though Nouha Dicko added a late consolation after Michael Duff saw red for the visitors.
Barnsley's win over promotion hopefuls Hull means Wolves must win at Brighton on the final day to stand any chance of survival.
Tongo Doumbia looked to have set the hosts' stall out early as he stung the palms of visiting goalkeeper Lee Grant inside the first five minutes. However, it was the Clarets who soon stole the initiative with Ings and Danny Lafferty both going close shortly after.
And it was Ings who was on hand to pounce after seven minutes, rifling in a fine strike from the edge of the box following a pin-point delivery from Kieran Trippier.
Sam Vokes came close to doubling the advantage but for a terrific stop from Dorus De Vries to keep out his volley as Wanderers continued to falter. With the tension inside Molineux almost tangible Dean Saunders' side finally rallied through Kevin Doyle, but he placed his effort straight into the gloves of Grant.
Matt Doherty had the luxury of a free header moments later but could only glance wide from Stephen Ward's cross to leave them trailing at the break.
The home supporters were demanding more after a lacklustre restart and those calls grew louder when they fell further behind. After a melee in the box Paterson popped up to slam the ball home on 53 minutes, adding to the already toxic atmosphere inside the ground.
With plenty of boos aimed at Saunders on the touchline the former Doncaster boss introduced attacking duo Bakary Sako and Dicko in hope of a response.
But it was the visitors who continued to threaten as Marney saw his effort well blocked by Doumbia before Vokes fired the rebound wide.
Wolves chairman Steve Morgan was next in the firing line with chants of 'You're taking us down' aimed towards the directors box.
They were offered a glimmer of hope shortly after the hour mark, though, when Duff was given his marching orders. The Clarets defender was shown his second yellow card of the game after a cynical block on striker Doyle after a mazy run towards goal.
Substitute Dicko gave them a lifeline two minutes from time when he poked home from 12 yards after a clever knock-down from Liam McAlinden.
But the visitors were not to be denied their first win at Molineux for almost six years as they saw off a spirited late fight-back.