Fresh from winning the Championship in 2017/2018, losing only one game in the process, Bristol capitalised on a couple of glaring errors from its opponent, which had won their opening game of the season in each of the past eight years.
After trailing 9-0 to a trio of Ian Madigan penalties, Bath briefly led thanks to Tom Homer's score — the full-back atoning for a dreadful mistake in the first half — and the boot of Rhys Priestland, who himself was far from at his best.
But in the final 15 minutes, Madigan slotted another penalty before Alapati Leiua crossed for what proved to be the decisive score, Madigan's missed conversion meaning Bath at least came away with a losing bonus point.
The tone for Bath's outing was set early on as Priestland struck the post with a penalty, sandwiched between two successful efforts from the boot of Madigan.
Worse was to come for the visitors, who saw Homer inexplicably knock on when putting the ball down for a simple score under no pressure, having been fed by Priestland, who appeared to be over the line in any event.
Madigan stretched Bristol's lead before Priestland's poor game continued as he sent a straightforward effort from the tee wide to the left.
Homer did not err with his handling in going over with half an hour to play and Tusi Pisi's yellow card then yielded three points for Bath, which went in front for the first time, only for Madigan and Leiua to ensure a successful return to the top flight for the Bears.