Josh Walker put the Owls in front on 34 minutes, only for Lewis Dennison to replay for the play-off chasing Villagers five minutes into the second half.
But the home side weren’t to be denied with Curtis Bateson bagging the winner for the leaders just before the hour.
“It’s a good win against good opposition and we’re delighted to get through it,” Cleethorpes boss Craig Rouse said.
“We’ve had illness ripping through the camp and a lot of them are out on their feet. They’ve put a massive shift in there.
“The message to the players now is just to look after themselves. It’s about resting up and making sure we get the players right for another big derby game at Grimsby Borough.”
Rouse’s men are being pushed all the way for the title by Belper Town, who rallied to a 4-1 win at bottom club Sheffield.
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