Steve Howard nodded the only goal early on before Rams boss Billy Davies was ordered to the stands in the 54th minute for arguing with referee Mike Riley.
The visitors piled on the pressure in the second half but lacked a killer touch in the final third.
And their misery was complete in stoppage time when midfielder Eric Djemba-Djemba and manager Steve Cotterill were dismissed.
But they made the worst possible start against a Derby side that was aiming for their fifth straight win, when Howard lost his marker and nodded home after just four minutes.
Rams midfielder David Jones swung a perfect free-kick on to the head of Howard, who made no mistake from four yards.
It was the ex-Luton man's 13th goal of an impressive debut season at Pride Park.
Howard's strike set the tone of the half and 15 minutes later Craig Fagan came within a whisker of doubling Derby's lead and opening his account following a transfer window move from Hull City.
The quick forward ran on to Stephen Pearson's defence-splitting pass and beat Michael Pollitt with a low effort - only to see the ball hit the inside of the post and bounce back into play.
Jones then tested Pollitt with a close-range header but the visitors finally got into the game and Derby keeper Stephen Bywater was forced to parry Joey Gudjonsson's swerving drive from 25 yards.
Burnley's improvement continued after the break and the leaders seemed tense in front of the huge home crowd before the dismissal of Davies threatened to further upset their rhythm.
The Scot was sent to the stands for arguing with referee Riley after the official, who had upset the Derby faithful with a string of contentious decisions all night, gave a foul against Fagan by the dug-outs.
But the visitors failed to seriously extend Bywater in the closing stages and Derby held on for an anxious - and potentially huge - three points.
In stoppage time, Djemba-Djemba was dismissed for his second booking before Cotterill was sent from the dug-out for clashing with Rams defender Mo Camara in the aftermath.