Sunderland took the lead inside four minutes when debutant Ellis Simms, on loan from Everton, opened the scoring.
The hosts drew level six minutes later through Andi Weimann, before Chris Martin put the Robins ahead.
Simms then added a second, allowing Ross Stewart to head home the winner 18 minutes from time.
Our chief Sunderland writer Phil Smith has dished out his player ratings from Ashton Gate:
Simms then added a second, allowing Ross Stewart to head home the winner 18 minutes from time.
5. Dennis Cirkin - 7
Came through a tricky first half to defend well and help get his side over the line. 7 Photo: Frank Reid
6. Jack Clarke - 8
What a start to the season he has had. Cleared one off the line in the first half and caused carnage down Bristol’s right flank in the second. Forced Nigel Pearson to change both his defenders on that side but it made no difference. 8 Photo: Martin Swinney
7. Corry Evans - 7
Took a yellow card in the first minute so did very well to manage that through the game. Looked a little off the pace in the first half but very, very good in the second. 7 Photo: Frank Reid
8. Dan Neil - 7
Struggled in the opening exchanges with some loose passes but settled and had some very good touches in the second half. Got on top of his duels. 7 Photo: Frank Reid