Ten men Mitres unable to complete second half comeback after poor first half.
It was a bottom two clash as Cleeve made the long trip to North Devon looking to close the gap on their hosts Barnstaple.
However it was the hosts who started the better putting the early pressure on with Jed Harper-Penman causing problems.
Harrison Iddles then broke through for Cleeve but looked to take one touch too many as the keeper came to collect the ball.
Barnstaple then started to take control of the half showing more desire and effectively sealed the match in a mad last 15 minutes of the half.
Tallan Mitchell put the hosts ahead on the half hour mark to the relief of the home faithful.
Then in a moment that shows how quickly football can turn Jack Watts saw his speculative effort spilled by the goalkeeper and just clawed off the line and Barnstaple went straight down the other end on the break to double their lead through Harry Stevens.
Chris Ibbeson made the task for Cleeve look almost insurmountable with a third for Barnstaple just before half time.
Things looked even worse for Cleeve when Ben Tunnicliff was sent off in an incident that set off a 21 man brawl.
As disappointing a performance as the first half was for Cleeve, the 2nd half was much better with more desire and flowing attacking moves shown.
Just 10 minutes into the half, Ieuan Crowe looped a ball into the top corner to give Cleeve a glimmer of hope but this seemed to really make the Barnstaple players nervous.
Cleeve soon pulled within 1 when Harlie Price hit a screamer into the top corner from 25 yds with over 25 minutes left on the clock.
Whilst Cleeve were on top they had to leave themselves short at the back and Barnstaple had a couple of counter attacks that nearly led to a goal, with one chance being cleared off the line.
As the game moved into added time the ball came out to Crowe on the edge of the box and he volleyed it straight towards goal only for the keeper to just tip it over the bar to leave Cleeve travelling back up the M5 10 points adrift at the bottom but with two games in hand still on Barnstaple.