The hosts were forced to come from behind, but hit back with tries from Harvey Russell, Joe Tucker, Alex Giltrow (2) and the returning Dean Ashfield.
The opening period was scrappy and disjointed as both sides set about running the Christmas break out of their legs.
The game was halted temporarily on 20 minutes following a serious looking injury to a Bristol Harlequins player.
The game was subsequently moved to the neighbouring pitch at the Vale to allow the injured player to undergo treatment – and when play restarted, Quins took the lead almost immediately - taking advantage of some sloppy defending to crash over in the corner.
For all their territory and possession, Clevedon once again found themselves with nothing to show for it – that was until Russell powered over from a driving maul to open the hosts’ account.
Clevedon didn’t have to wait long for a second, as Joe Tucker burst down the right-hand touchline before darting in under the posts to give his side a half-time lead.
The second half began in a similar style to the first, with high error counts from both sides.
Quins enjoyed lengthy spells of possession, but a robust home defence kept them at bay.
The shackles were broken on 65 minutes when Giltrow barrelled his way over from close range – and Dean Ashfield marked his return from a long-term knee injury with Clevedon’s fourth, latching on to a pinpoint pass from Tom Francis to touch down.
A fine blindside break from Mike Lowis rounded off proceedings, as he fed Giltrow for his second and Clevedon’s fifth with the final play of the game.
Clevedon return to league action next Saturday at home to Camborne, kick-off is 2.30pm.