A poor recent run had dropped the Steelmen out of the top six race but two wins this week has lifted them to within five points of sixth-placed Chasetown, who have played two fewer games though.
Corby raced into a 3-0 lead inside the first 28 minutes with goals from Steven Leslie after six minutes, Jake Bettles after 25 and Sam Warburton three minutes later.
The Steelmen made it game over with number four in the 55th minute through Phil Trainer before Cleethorpes pulled one back four minutes from time through Danny North.
Thursday`s other game saw Carlton Town and Peterborough Sports – two teams still not entirely safe from finishing bottom – share a 1-1 draw at the Bill Stokeld Stadium.
Josh Moreman put the visitors ahead after 14 minutes and the Nottinghamshire side pulled level through Ben Hutchinson right on the stroke of half-time.
Image courtesy of @CleeTownFC/Twitter
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