Jay Rodriguez scored the winning goal in the final minute of extra-time to book Burnley's place in the fourth round of the FA Cup.
The Clarets now face a trip to West Brom following the Baggies' 2-0 win over Peterborough.
A dire first 45 minute
s was followed by two goals in the second period.
Samuel di Carmine curled home the opener for the visitors but Steven Thompson's thumping equaliser from close range meant an extra 30 minutes was needed.
And after a minute of added time in injury time local lad Rodriguez raced through and notched the decisive goal.
Only a Steven Caldwell interception prevented a fine passing move from the visitors resulting in the opener.
Midfielder Emmanuel Ledesma threaded a neat ball through but a sliding challenge from the Clarets defender took the ball away from an advancing Di Carmine.
Alan Mahon's left-footed drive was the closest either side came to breaking the deadlock.
The visitors scored in the 54th minute when Mikele Leigertwood stole the ball from Burnley substitute Mahon and sent Di Carmine clean through on goal.
The Rangers frontman drove towards goal before curling his first goal in three months into the far corner of Jensen's net.
And it could have been two when midfielder Martin Rowlands hit the crossbar twices, while Chris Eagles was denied a stoppage-time winner after he was ruled to have been offside when he turned Alex McDonald's drive into the net.
Heidar Helguson was denied a goal by a late offside flag in the 103rd minute for Rangers.
Route one football resulted in the winning goal as Brian Jensen's long kick left Rodriguez with just Cerny to beat and he made no mistake with his finish.
Burnley: Jensen, Alexander, Carlisle, Caldwell, Jordan, Elliott, McCann, Gudjonsson, Eagles, Blake, Thompson. Substitutes: Penny, Kalvenes, Akinbiyi, Mahon, Rodriguez, MacDonald, Kay.
QPR: Cerny, Hall, Stewart, Gorkss, Delaney, Alberti, Leigertwood, Mahon, Rowlands, Ledesma, Di Carmine. Substitutes: Borrowdale, Crowther, Rose, Connolly, Ephraim, Helguson, Ramage.
Referee: C Webster (Tyne & Wear).
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