Two tries in the last 10 minutes turned Barnstaple from likely losers into 31-19 winners at Cleve.
The Bristol side were 14-5 up at the break and 19-5 in front with 50 minutes played.
At that stage in the game Cleve had tries on the board from scrum-half Josh Mortimore, centre Harry Owens and Nathan Huntley. Lachlan Kirk kicked two from three.
Barnstaple’s only score before the break was a try by wing Ryan Lee.
The revival started with a second try from Lee, then one from lock Adam Lloyd that Jake Murphy converted to leave two points in it.
Full-back Kryan McManus was Barum’s match winner with tries on 70 and 76 minutes, both converted by Murphy, to settle the game.
Victory took Barnstaple up to third in the table. Maidenhead dropped down to fourth after they lost at Exmouth.
Barnstaple’s aim this season is to get straight back into National Two South at the first attempt.
Jeremy Chugg, the director of rugby at Pottington, said with five wins in six starts, the promotion train is on schedule.
“Even at this early stage you can see three or four teams pulling away at the top and I don’t think that group will change much,” said Chugg.
“Bournemouth and Ivybridge have not lost yet, we are right behind them and Maidenhead just behind us with Exmouth and Exeter University.
“We always back ourselves to win our home games and would be disappointed if we lost more than a couple all season. The other teams around us would probably say the same.
“If we want to go up we can only afford to lose three of four away from home.
“The way I look at it our results against the other top-four, top-five sides are going to decide how we do.
“If we can nick a win here and there off one of the others, that is going to improve our chances.”