The in-form Rams took the lead through Luke Moore's deflected effort and, considering the number of chances the hosts failed to convert, it looked like being a frustrating night for Brian Laws' men.
However, Tyrone Mears' first league goal in two-and-a-half years against his former club levelled things up.
And Chelsea loanee Jack Cork's flicked header settled a keenly-fought contest on a freezing night at Turf Moor.
The Clarets made a bright start but Chris Iwelumo will have been disappointed after failing to direct Ross Wallace's left wing cross towards goal.
A well-timed pass from Robbie Savage in the 11th minute created Derby's first opening for Moore, but Brian Easton tracked back and produced a crucial tackle to deny him a chance at goal.
Jay Rodriguez's through-ball had just too much on it for the fast advancing Dean Marney minutes later and it ran through for Rams stopper Frank Fielding.
Andre Bikey shook off the attention of Shaun Barker in the 18th minute, but could only head down into the arms of Fielding as Marney's corner dropped perfectly in his path.
The visiting Rams then went ahead when Moore scored his third of the season with 19 minutes gone.
Moore, on loan from Aston Villa, neatly chested the ball down, cut inside Bikey and unleashed a ferocious shot towards goal from 25 yards.
The ball took a sizeable deflection off the outstretched leg of Clarke Carlisle and Grant, who had already moved to his left, could only watch as it finished up just inside his right-hand post.
As Burnley searched for an equaliser, Rodriguez forced a smart stop from Fielding in the 31st minute with a shot on the turn at the near post.
Chances were at a premium, though, as the visitors continued to hold firm, taking a one-goal lead into the interval.
Derby took the ball to the corner flag with just five second-half minutes gone, signalling their intention to defend their slender lead.
Marney, who tried his luck with a number of long-range efforts, did so once again but this time it rippled the side-netting.
Rodriguez fired a tame effort towards Fielding's goal, saw penalty claims waved away and then crossed for Iwelumo, who could only find the roof of the net, as he looked to spark the Clarets back to life.
Bikey then felt he had been held by Barker in the box but referee Mark Haywood thought otherwise.
It was all Burnley and Wallace's snapshot clipped the top of Fielding's crossbar in the 66th minute with the Rams goalkeeper well beaten.
The equaliser, which was always coming, arrived seven minutes from time when Mears curled home a superb free-kick.
Dean Moxey was the man to concede it, halting Marney in full flow.
After a lengthy delay, Mears angled the resulting set-piece over the wall and into the bottom left-hand corner of Fielding's goal.
Cork, who had been denied minutes earlier by wonderful Fielding save, then popped up at the front post in time added on to glance home the winner from Wade Elliott's cross.
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