Bristol took the lead with tries from James Williams, Ellis Genge and Gabriel Ibitoye, with Quins cancelling out the first two through Chandler Cunningham-South and Alex Dombrandt.
After Harry Thacker extended Bristol's lead to 12 points with a try early in the second-half, Quins went ahead for the first time with a Luke Northmore score and Cunningham-South's second.
But Williams knocked over two penalties and converted tries from Max Lahiff, Noah Heward and Gabriel Oghre to give the Bears a handsome winning margin and secure a fifth-place finish.
Bristol were at their free-flowing best in attack in a thrilling first half, with excellent handling opportunistic off from loads combining with an ambition to move the ball wide as quickly as possible that left Quins scrambling in defence.
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