Jeff Hendrick volleyed Nigel Clough's side into the lead in the 34th minute before Craig Bryson nodded home a second to delight the 846 travelling fans.
The Rams had goalkeeper Adam Legzdins to thank for keeping their two-goal lead intact though, pulling off a string of fine stops to help his side to a rare away victory.
Sam Baldock's two goals helped City to record a vital victory at Hillsborough last weekend and a fine save from Legzdins stopped him from scoring yet again after Ryan Taylor picked out the former MK Dons striker.
The Robins had conceded more goals than any other side in the division coming into this match and they let in their 44th of the campaign in the 34th minute when Jacobs' cross found Hendrick on the edge of the six-yard box.
And just two minutes later that lead was doubled when Gareth Roberts swung a ball in from the left wing that Bryson nodded past Tom Heaton from five yards out.
City almost found an instant response just seconds later when Albert Adomah rampaged forward and set up Baldock but his effort was kept out by the quick reactions of Legzdins.
Derek McInnes shuffled his pack at half-time with Taylor being replaced by top scorer Steve Davies as they searched for their first home win since August 25, a run stretching back eight games.
But the visitors, who came into this clash just three points off a play-off place, had the first real chance of the second half when Jacobs' effort from the edge of the box was deflected wide by Stephen McManus.
Just two minutes later Davies headed Adomah's corner wide from 12 yards out and in the 74th minute the Scot's came close to his ninth of the season with a 20-yard curling effort which was saved well by Legzdins.
City were really upping the ante and had another two chances in quick succession. First Baldock fired over the bar before Adomah's looping cross found defender Liam Fontaine and his close-range header forced Legzdins into a superb save.
Bristol City striker Jon Stead made a welcome return from injury for the Robins but his only real chance, a long range effort with five minutes to go, was comfortably saved by the Derby keeper. And Legzdins was called into action once again minutes later to push away Adomah's volley.
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