The visitors weathered early pressure before taking a 34th minute lead through a Jean Louis Valois free-kick and then held out until 20 minutes from the end.
Derby, without being at their most fluent, had pushed Burnley back and got their rewards when Peschisolido netted the rebound after Brian Jensen had blocked Adam Bolder's low shot.
The scoreline was just about right on a bitterly cold night with a snowstorm swirling around Pride Park for much of the second half.
But despite the conditions, both teams tried to pass the ball and Burnley's solid and well-organised approach certainly gave Derby problems.
But Jensen still needed to be at his most alert to keep out a 25-yard free-kick from Inigo Idiakez and a snap-shot from Grzegorz Rasiak in the opening 17 minutes.
Burnley created few clear chances for all their neat passing but were unlucky when a Micah Hyde header struck the post and hit Bolder in the back before bouncing into the arms of Lee Camp.
But there was no escape for Derby when Tom Huddlestone fouled Ian Moore and Valois bent a free-kick over the wall and in off Camp's left-hand post.
Derby responded with some concerted pressure in the second half and Idiakez was unlucky when a 20-yard shot came back off a post.
Jensen also denied Peschisolido but the Rams at last broke through in the 70th minute.
Rasiak found Bolder on the left, and although Jensen beat away his low shot, Peschisolido was on hand at the back post to slide the loose ball into the unguarded net.