The visitors nearly took the lead in the first minute when Kyle Parker broke down the left but his shot skewed wide.
Blaine Waugh then shot wide for the hosts before Fairford ‘keeper Sam Ayres took out Jake Rhodes in the box. Jack Watts stepped up to convert the resulting spot kick.
Archie Haskayne and Kieran Alder both saw shots blocked before Rhodes saw his freekick beat Ayres but hit the post.
With Cleeve in control the task looked even tougher for Fairford when George Peare lunged in on Harrison Reeves from behind. With Reeves brushing the challenge off the fans in the stand thought the yellow was coming but instead it was a straight red for the Fairford man.
However, after the interval it was the ten men of the visitors who looked more likely to score, putting pressure on Cleeve and Dan Bailey’s driving run and shot forced a fine save from Sam Gilder.
Danny Gudger then saw two decent efforts go close and Ross Langworthy’s lob went wide after a defensive mix up as the visitors looked to capitalise on their good spell, however the Cleeve side defended well all over the pitch.
As the game went into added time Cleeve looked more likely to score and Kevin Slack looked through from a Gilder kick with Jacob Humphries hauling him back, but not enough to stop Slack latching onto the ball and converting. However, the whistle had already blown as with Humphries being last man he was shown a straight red.
Cleeve’s next match is at home to Westbury United on Saturday.