Goals in either half from Danny Ings and Martin Paterson were enough to see Burnley 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, but it was the Clarets who soon stole the initiative with Danny 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.
With the tension inside Molineux almost tangible, Saunders' side finally rallied through Kevin Doyle, but he placed his effort straight into the gloves of Grant.
The home supporters were demanding more after the 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 after 53 minutes, adding to the already toxic atmosphere inside the ground.
The Wolves fans were offered a glimmer of hope shortly after the hour mark when Duff 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 first win at Molineux for almost six years as they saw off a spirited late fightback.