Steve Howard scored his 12th goal of the season to see off 10-man Burnley and extend Derby's lead at the top of the Championship table to six points.
The winner came as early as the fifth minute, Howard heading home with aplomb from David Jones' pin-point free-kick.
The striker also hit a post, while Joey Gudjonsson came closest for Burnley.
However, the tie ended in controversial fashion with both managers dismissed for arguing and Eric Djemba-Djemba sent off after collecting two yellow cards.
"As for why (manager) Billy Davies was sent off, I don't know why - he asked me to come and face the press.
"I guess he thought it was a bit petty but it's probably a bit early to say if we will appeal. If you're not passionate in these games you don't deserve to be there."
"And, to make matters worse, I get sent off for asking Mo Camara why he pushed Kyle Lafferty. They'll say somewhere along the line I used some language - no chance.
"Do we take the passion out of the game - because if we do the fans'll stop coming? Did Djemba-Djemba deserve to be sent off? No. Did Billy Davies? No."
Derby: Bywater, Edworthy, Leacock, Moore, Camara, Fagan, Oakley, Pearson, Jones, Teale (Bisgaard 61), Howard.
Subs Not Used: Michael Johnson, Peschisolido, Lupoli, Malcolm.
Booked: Oakley, Fagan.
Goals: Howard 4.
Burnley: Pollitt, Thomas (Foster 45), McGreal, Duff, Harley (McCann 88), Elliott (Mahon 70), James O'Connor, Gudjonsson, Djemba-Djemba, Akinbiyi, Lafferty.
Subs Not Used: Coyne, Garreth O'Connor.
Sent Off: Djemba-Djemba (90).
Booked: Mahon, Djemba-Djemba, Foster, Harley.
Att: 23,122
Ref: M Riley (W Yorkshire).