Three penalties from Mike Taylor were all the hosts had to show for long periods of possession and territory – although they went close to snatching the win with the final play, but winger Joe Tucker was unable to touch down as the covering tackle came in.
Clevedon started brightly and went ahead courtesy of Taylor’s boot on ten minutes.
They doubled their lead shortly after as Taylor was once again on target from in front of the posts.
Chris Warren threatened the visitor's defence on a couple of occasions with some elusive running, but Clevedon were unable to apply the finishing touches when in good positions.
Drybrook hit back with a penalty of their own on the stroke of half-time to send the sides into the break with the score at 6-3.
After the break, Harley Foley - who impressed throughout the 80 minutes - went close to extending the lead further following a strong carry, but the ball was dislodged five metres from the line and the chance went begging.
Drybrook soon began to find their feet in the game and following a prolonged period of possession, touched down in the corner with just 12 minutes to play – a fine touchline conversion gave them the narrowest of leads heading into the final stages.
Taylor was off target with a tough penalty from the left-hand touchline, before Drybrook put daylight between the sides with a well-struck effort from 30 metres.
With seconds remaining, Tucker picked up from a ruck and drove powerfully towards the line – but the covering defence scrambled their way across to force the knock-on and condemn Clevedon to an agonising defeat.
The defeat sees Clevedon slip to the bottom of the table, but that could all change next weekend as they travel to second-from-bottom Matson.