Cleethorpes Marc Cooper missed an early penalty and the Owls were made to pay as Kyle Bensley gave the home side the lead with his own spot kick.
Darren Hanslip equalised just before half time and the Owls sealed the win with second half goals from Danny Trott and Lawrence Heward. Boro were reduced to ten men near the end as former Owl Jason Flint saw red for a headbutting incident.