The Lancashire club had not won at Oakwell for 70 years but they embarked on a bold second-half comeback last night after Barnsley had gone three goals ahead.
Muller, however, thwarted Chris McCann and Chris Eagles in turn near the end to ensure B
arnsley secured victory.
The result was shrouded in controversy as television replays proved that Barnsley's vital second goal should have been ruled out for offside.
Winger Simon Whaley was standing on the wrong side of the Burnley defence when a shot from Jamal Campbell-Ryce deflected off his back and into the net.
Burnley manager Owen Coyle was still incensed with referee Nigel Miller at the final whistle and went onto the pitch to vent his disapproval.
A goal from striker Jon Macken had put Barnsley ahead on 18 minutes before Whaley added the second just after the break.
Moments later a shot from substitute Diego Leon deflected off Burnley defender Steven Caldwell to make it 3-0 and seemingly put Barnsley in control.
But Burnley substitute Martin Paterson scored two goals in four minutes to set up a tense finale.
The victory lifted Barnsley, who were in the relegation zone not long ago, to within four points of the play-off places.
With only two defeats in their last 19 games (against Reading and Wolves), Burnley had arrived at Oakwell as one of the form teams in the Championship.
The Clarets had also joined Barnsley in that elite group of clubs able to boast a 2008 victory over Chelsea after a penalty shoot-out success in the last round of the Carling Cup.
Barnsley, who beat Chelsea on route to last season's FA Cup semi-finals, took the lead against the run of play.
Rob Kozluk's through ball caught the visitors flat-footed at the back and left striker Macken unmarked to convert his fifth goal of the season.
Burnley had looked the more likely scorers until that point, playing an expansive passing game which had Ade Akinbiyi in front, and Graham Alexander behind, in a creative four-man line.
Muller had pushed a low drive from Eagles around the post in the first half and Macken had been denied by Brian Jensen.
Barnsley certainly missed the suspended Anderson De Silva in the centre of park. His partnership with the Argentinian Hugo Colace has been central to the club's recent climb up the table.
However, despite spending more time defending than attacking, Barnsley could still have doubled their advantage before the break.
Kayode Odejayi muscled his way past Clarke Carlisle into a one-on-one situation with Jensen but could only shoot straight at the goalkeeper's legs.
It was the kind of miss that makes Iain Hume's absence even more regretable for Barnsley. Striker Hume is out for the rest of the season after suffering a fractured skull against Sheffield United this month and home supporters made a point of chanting his name in the opening minutes of the televised game.
Barnsley's rapid fire double salvo, at the start of the second half, relied heavily on a mixture of both luck and poor refereeing but Burnley shrugged off the setback to mount a brave revival. Substitute Paterson started the comeback when running onto a pass and turning the ball past Muller.
Just moments later, the Northern Ireland international reduced the deficit to a single goal, seizing on a loose ball from Kozluk then embarking on a dash into the area,where he produced a neat finish.
Barnsley managed to hold onto the three points but only after the two vital late saves from Muller.
After the game Barnsley manager Simon Davey revealed that the Football Association will make a decision this morning whether to take action against Sheffield United defender Chris Morgan for his challenge which left striker Hume with a fractured skull.
Barnsley: Muller; Hassell, Souza, Foster, Kozluk; Whalley, Campbell-Ryce, Colace, Devaney (Leon 11 (Van Homoet 85)); Macken, Odejayi (Coulson 85). Unused substitutes: Mostto, Rigters.
Burnley: Jensen; Duff (Rodriguez 88), Carlisle, Caldwell, Jordan; Alexander; Eagles, McCann, Gudjonsson (McDonald 55), Blake; Akinbiyi (Paterson 55). Unused substitutes: Penny, Mahon.
Referee: NS Miller (County Durham).
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