Harley Foley, Owen Biggins and Aaron Rice all crossed for second half tries after the home side had taken a 7-0 lead into the interval, and Clevedon led 19-10 with 70 minutes on the clock
But Weston rallied in the final ten minutes, scoring twice to deny the visitors a sixth successive league win.
In front of a bumper crowd at the Recreation Ground, it was the home side who were quickest out of the blocks, and after fly-half Dan Lomax had been wide with a penalty kick at goal, wing Matt Watkins scorched over the open the scoring.
Clevedon gradually fought their way back into the contest, but failed to turn possession and territory into points and were forced to settle for a seven-point half-time deficit.
That all changed after the break, as the away side hit top gear in a scintillating 20 minute passage, crossing three times to turn the game on its head.
First Harley Foley applied the finishing touches to a sweeping handling move, after Brian Largenton and Gary Carpenter had burst through the hosts' midfield - and then Owen Biggins crashed over from close range, with Gary Carpenter again involved in the build up.
Lomax dissected the posts with a penalty on 55 minutes, before Aaron Rice sprinted in for Clevedon's third score on the hour mark, after neat hands from Matt Honour and Brian Largenton.
Will Carpenter converted two of the three as Clevedon opened up a nine point lead heading into the final ten minutes.
But Weston were far from finished and after Watkins had touched down for his second of the evening, the referee awarded the hosts a penalty try on 78 minutes, which Lomax converted to condemn Clevedon to a 22-19 defeat.
Clevedon kick-off 2016 with a home game against Drybrook at Coleridge Vale - 2.30pm KO.