A penalty from Mike Taylor accounted for the home side's only points of the afternoon, despite having long periods of possession and territory.
This was a game Clevedon could and probably should have won, but they were eventually made to pay by a far more clinical opponent.
Tries from James Bryant, and Luca Ercolani sealed the win for Bridgwater, while Chris Ashwin kicked eight points.
Bridgwater took a lead early when Clevedon were offside in midfield, with Chris Ashwin landing the straightforward penalty kick.
Clevedon hit back and surged into the Bridgwater 22 as they began to gain a foothold - and Mike Taylor was just wide with a 28m penalty from near the right touch line.
Wing Oscar James then demonstrated pace and power to break through to close to the Bridgwater line – and when fly half, Ashwin infringed in the red zone, he became the first player to be yellow carded.
Taylor knocked over the simple penalty from in front of the posts to level the scores.
Despite being a man down, Bridgwater scored their first try of the afternoon to go 8-3 ahead, as James Bryant crashed over from a driving maul.
On the stroke of half time, Taylor attempted penalty from just inside his own half but was narrowly short.
After the break, Clevedon dominated the early exchanges but once again had nothing to show for it.
Largenton, Taylor and James all went close to crossing for Clevedon’s opening try, but all too often the final pass went astray with the line begging.
Bridgwater clawed their way back into the game and began to exert pressure – and this pressure led to Clevedon centre Aaron Rice being sent to the bin.
The resulting penalty was kicked to the corner and when Bridgwater set up the inevitable driving maul, replacement flanker Ercolani, powered over for the try.
But 14-man Clevedon continued to attack and after a scything break from Joe Tucker saw the replacement held up agonisingly short of the line, Bridgwater’s Gavin Knight joined Rice in the sin-bin.
Ercolani soon became the third man to take a ten minute break as Clevedon piled on the pressure, but Bridgwater’s defence stood firm.
When Clevedon did eventually get over the line, they could not ground the ball as Rice was held up in goal – and with time ticking away, the visitors held on to take victory.
Clevedon will hoping to return to winning ways this weekend when they travel to Newent.