Beaten at Leicester and Nottingham Forest in their previous two away matches, the Clarets twice fell behind to goals from Theo Robinson and Jamie Ward.
Chris Eagles equalised on each occasion - the second came on the rebound after his penalty was saved by Brad Jones following Ward's sending-off - before providing the cross for Wade Elliott to head Burnley in front.
Chris McCann, starting his first game of the season following reconstructive knee ligament surgery last summer, sealed the win midway through the second half as Eddie Howe's side moved level on points with six-placed Forest.
Derby defender Russell Anderson made his first start since the opening day of the season in the only change made by Rams boss Nigel Clough. The Scot replaced the ill Gareth Roberts.
Burnley also made a solitary change as McCann returned. The midfielder, who had previously played only 100 minutes since September 2009, came in for the injured Dean Marney.
Marney took a knock in the midweek victory over Middlesbrough and, unlike Andre Bikey and Eagles who limped off in the second half, failed to recover in time.
Derby took the lead after six minutes when Robinson caught the Burnley defence napping and nipped in the fire Stephen Pearson's left-wing cross past the helpless Brian Jensen.
Eagles equalised after 15 minutes with an audacious chip from the edge of the box that Jones could only watch drift into his net.
There were chances at either end as Jay Rodriguez and Robinson went close before Jones tipped Elliott's shot round a post midway through the first half.
Derby regained the lead just after the half-hour mark. Ward claimed the goal but his 20-yard shot was helped past Jensen with the aid of a big deflection.
Steve Davies went close to extending the Rams' lead before the break, while McCann almost equalised early in the second half.
Burnley were level after 62 minutes when Ward was sent off for a foul on former Derby defender Tyrone Mears. Jones saved Eagles' spot-kick but the Clarets man tucked home the rebound.
Mears, booed throughout by the home fans on his return to Pride Park, was involved again as Burnley edged ahead three minutes later.
He set Eagles on his way and Elliott headed home the cross from 12 yards out.
McCann gave the visitors some breathing space with 16 minutes remaining when his 25-yard shot struck the inside of the post and nestled in the net.