Derby started the first half impressively and enjoyed a healthy spell of possession in the opening five minutes.
However, it was Burnley who nearly took the lead from a free kick, with Ben Mee heading wide from 12 yards out.
It didn't take long for the visitors to create another chance and this time it was taken as Danny Ings started an elaborate run from midfield, which went through the Derby defence, presenting him with the chance to fire low down past goalkeeper Lee Grant.
Burnley went 2-0 ahead following poor Derby defending which saw them unable to deal with a cross into the box, leaving Sam Vokes with the chance to turn the ball past Grant in the Derby goal.
County almost grabbed one back on the stroke of half time, with Johnny Russell doing well to find room in the box, but his snap-shot flashed wide.