Victory for the hosts was all the sweeter given Bristol's humiliating 94-33 defeat by league leaders Northampton last week. It meant that reigning champions Bath are still looking for their first Prem win in Bristol since September 2008.
This result left Bath in second place, with Bristol fifth and on the verge of the title-deciding play-offs ahead of the bulk of the weekend's fixtures.
Bristol's tries came from Fitz Harding, Max Lahiff and a penalty try, with both AJ MacGinty and Tom Jordan adding conversions.
Ollie Lawrence, Guy Pepper and Thomas du Toit scored Bath's tries. Ciaran Donoghue kicked two conversions, with the title-holders 19-7 ahead at the break.