An early goal from Colin Kazim-Richards, his fifth of the season, was enough to secure the points although the victory was marred by what appeared to be a serious injury to Polish midfielder Krystian Bielik.
Bielik, who has been a key figure in Derby's recent revival under Rooney, was taken away on a stretcher in the first half.
City had a chance to equalise through Famara Diedhiou but created few chances and Derby were good value for another win, leaving them five points above the drop zone.
Derby had scored only six goals in 13 home games but they added to that in the opening five minutes.
Lee Buchanan delivered a superb cross from the left in the fourth minute and Kazim-Richards arrived at the far post to fire high into the roof of the net.
City had been pushed back by Derby's positive start and they were fortunate not to concede again when Max Bird's shot from the edge of the box rebounded off a post in the 15th minute.
City had a chance to regroup when there was a lengthy stoppage after Bielik, who missed most of last season with a knee injury, went over near the right touchline and was stretchered off in the 36th minute.
It was much better from City who ended the half on the front foot with Alfie Mawson heading at Marshall in added time.
City had a good chance in the 51st minute when Jack Hunt crossed from the right but Diedhiou failed to get decent contact on his header.
Bentley was lucky when he misjudged a cross in the swirling wind but recovered to scramble the ball clear before Derby could pounce.
The keeper escaped again in the 74th minute when he dropped a corner but Adrian Mariappa hooked the ball off the line.
Bentley redeemed himself four minutes later when he kept City in the game by turning over a deflected shot from Martyn Waghorn.
The visitors had been better since the break but had not seriously tested Marshall until a free-kick fell to Mawson whose low shot forced the Scotland international into a diving save in the 85th minute.