Regular season table-toppers Bath, who've already won the European Challenge Cup this season, were 13-6 behind at half-time, with Bristol leading through two AJ MacGinty penalties and a try from James Dun.
All Bath had to show for their efforts in the opening 40 minutes at their Recreation Ground home were two penalties from Scotland fly-half Finn Russell.
But they went ahead early in the second half as Ted Hill and Joe Cokanasiga scored tries before Will Muir and Max Ojomoh crossed Bristol's line to leave the Bears with a gap they never looked like overcoming despite Bernhard Janse van Rensburg pulling a try back.
Bath, beaten in last season's Premiership final by Northampton, will face either Leicester or Sale -- who play on Saturday -- at Twickenham on June 14 as they bid to win their first league title since 1996.