The build-up to the match had been focused on talk of a merger between the two clubs next season, a scenario which would spare Bath relegation, as they currently sit three points adrift at the foot of the table.
Bristol's win in front of over 20,000 supporters – the biggest ever Premiership crowd – leaves them two places clear of the drop zone, four points ahead of Bath, having played the same number of games.
London Irish lie second-bottom – three points clear of Bath – after beating champions Leicester on Saturday, leaving Bath staring down the barrel of relegation.
In the end, it was a late interception try from Argentina fly-half Felipe Contepomi, which broke the hearts of the Bath players and fans, a late try from Bath's Ireland centre Kevin Maggs not enough to claim even a bonus point as Bristol twisted the knife with an injury-time drop-goal from Australian Shane Drahm.
Earlier Drahm had kicked Bristol into the lead with an 11th-minute penalty, only for Wales back rower Gavin Thomas to give Bath a glimpse of hope with a try nine minutes later, converted by Olly Barkley.
Barkley nudged his side further ahead with a penalty on the half-hour, but Bristol's former All Black centre Daryl Gibson gave the capacity crowd something to shout about when he picked up the ball after a break by Drahm to cross the line for their first try, converted by Drahm.
While fans in and around the stadium continued to protest against a possible merger between these two sides, the players continued the on-field drama, Barkley nudging over a second penalty for the visitors just before the interval to tie the score at 13-13.
It was however to be another 40 minutes before Bath scored again, and by then the Shoguns had amassed two converted tries through Contepomi.
Maggs hit back late with a try for Bath, but Drahm's drop-goal ensured that the visitors took nothing away from the game.
Where now for West Country rugby in the Premiership?
For Bristol Shoguns:Tries: Contepomi 2, GibsonCons: Drahm 3Pens: Drahm 2DG: Drahm
For Bath:Tries: Thomas, MaggsCons: Barkley 2Pens: Barkley 2