Bath, who finished 11 points clear at the top of the table, remain in the hunt for a treble having already won the Premiership Rugby Cup and European Challenge Cup.
The hosts secured a home semi-final with three rounds still to play and rested all of their first-choice players against Saracens a week ago, the same day Bristol's play-off fate was confirmed.
The Bears conceded they were underdogs coming into the match yet had the upper hand in terms of head-to-heads, having won five of their past six Premiership meetings.
If there was any psychological advantage to be had, they tried to seize it early on.
Bristol stuck first through MacGinty's penalty and as they have all season looked threatening whenever they quickly moved the ball.
Dun ran in next to the posts for a breathtaking try that began five metres out of Bath's try-line after the hosts kicked the ball deep into Bristol territory.
Kalaveti Ravouvou won the foot race with Muir, collected the ball and made 40 metres solo, skipping through one tackle and bouncing off another.
Janse van Rensburg's clear-out opened space for Ellis Genge to pass to Bath-born Dun to make it 10-0.
Indiscipline has been a constant issue however for the Bears and they were needlessly on the backfoot as Ravouvou tripped Ben Spencer and was sent to the sin-bin, to allow Russell to kick their first three points.
Still Bristol had Bath on the ropes at both ends. Some incredible ball-handling almost saw them score again only to lose it inches from the line, while at the other end full-back Noah Heward turned Bath over to help deny them any tries in the first 40 minutes.
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