Louis Grant and Luke Mascall put Cleethorpes into a 2-0 lead before the hosts pulled a goal back and the Owls were reduced to ten men when Laurence McKay was sent off.
The Owls are in NCEL League Cup semi final action on Thursday when the team travel to play Tadcaster Albion. The match will kick off at 7.45pm.
The teams will meet again just two days later when the final NCEL home match of the season takes place on Saturday as Tadcaster visit Bradley. Kick off will be at 3pm.