Former Rams midfielder Jones struck in the first half before Marney doubled the lead in the second, while Derby - now five points behind the Clarets in third - had Chris Martin controversially sent off just before half-time.
Martin received a second booking for a dive in the penalty box, much to the anger of the Derby dugout.
Burnley's Danny Ings almost sparked the game into life in the sixth minute but, after carrying the ball from just inside the Derby half, he dragged his shot wide.
Ings then turned provider on 13 minutes, jinxing inside and feeding strike partner Sam Vokes, but his shot was well blocked by Jake Buxton.
Buxton was the Rams' saviour again just before the half-hour mark, sliding in to thwart Vokes after a pinpoint cross from Michael Kightly.
But the Clarets did take the lead a minute later, Jones giving Lee Grant no chance with a superb half-volley after controlling a chipped pass from Ings.
Derby were reduced to 10 men just before the break, Martin picking up his second yellow card for a perceived dive in the Burnley box having earlier been booked for a clash with Jason Shackell.
It looked harsh on the striker, who appeared to lose his footing rather than go actively looking for the spot-kick, and Rams boss Steve McClaren made his feelings clear to referee Robert Madley at the break.
To their credit, the visitors came out strongly in the second half and a relieved Tom Heaton gathered substitute Conor Sammon's low shot at the second attempt just three minutes in.
On-loan Chelsea youngster Patrick Bamford also made an impact after coming on, twice going close with efforts from long range.
But after being under pressure for large spells of the second half, Burnley doubled their lead after 68 minutes with Marney prodding home from close range after Derby failed to clear a corner.
Bamford continued to be the main threat for the third-placed Rams and he felt he should have had a penalty on 78 minutes after going down under pressure from Shackell.
Those appeals were waved away by referee Madley, much to the youngster's annoyance, but he dusted himself down to feed Craig Forsyth, whose shot to the near post was well-saved by Heaton.
Marney lashed a wild effort well over the bar late on as the Clarets looked to increase their advantage, while Vokes fired in much the same direction from close range after a neat cutback from Ross Wallace.
Derby continued to press but were thwarted by a stubborn Burnley defence who collected another clean sheet.