The visitors outscored Albion by four tries to one to earn a third away win of the season and extend their unbeaten run to eight games in all competitions.
Dean Ashfield, Harley Foley, Chris Warren and Tom Thie all crossed for Clevedon to secure a vital bonus-point in their bid to beat the drop, while Mike Taylor kicked a conversion and a penalty.
Playing against a stiff downfield breeze, Clevedon started as they had finished at Bideford, playing with pace and intensity.
It didn’t take long for Dean Ashfield to open the scoring, as he ghosted through a hole in midfield to touch down under the posts.
Minutes later, Clevedon doubled their lead when man of the match Ashley Vailes intercepted on half-way, before feeding the supporting Chris Warren to dive over in the corner.
Bridgwater opened their account on the 20 minute mark courtesy of a Stuart Heal penalty, but Clevedon hit back almost immediately with Harley Foley rounding off a neat handling move on the short side to score his side’s third.
Early in the second-half, Taylor slotted a 30 metre penalty to extend the lead to 15 points – but Albion soon crossed for their opening try of the afternoon, with hooker Matt Hastie getting on the end of a sweeping handling move to give his the hosts a lifeline.
But that was as good as it got for Bridgwater - and it was one of their former players who put the game to bed on the hour, as Tom Thie scooped up a loose ball to touch down in the corner, with Taylor adding the conversion.
Another former Bridgwater player, Owen Biggins, ensured the Clevedon line would not be breached again with an expert turnover close to his own line – allowing brother, Rory to kick to touch and bring an end to proceedings.
Clevedon welcome North Petherton next weekend – kick-off is 2.30pm.