Graham Nicholas was forced into more changes for Towns' first ever trip to the catchily named Bradley Football Development Centre.
Max Newbould continued between the sticks in place of Brazilian based Ben Hunter, but a nasty collision for the Owls' fourth goal put him out of action for the second half.
Nick Black, Jay Davis, Andrew hawksworth, Tim Robertson, Sam Akeroyd, Sinmi Oyebanji, Curtly Martin Wyatt and Danny Harris, were some of the players ruled out, while John Hubball, Joel Barnett and Connor Johnson were brought in for first starts.
Cleethorpes threat was imminent, when Louis Grant struck a post within three minutes. The lively forward didn't need to wait long before chalking up his first goal, lashing home from close range after Newbould had saved from Alex Flett.
Tawanda Rupere went close for Garforth before Grant doubled the Owls' lead. This time guiding a lovely left foot effort beyond Newboulds' reach.
The Miners' new look side barely had time to gel before Cleethorpes effectively killed the game off within thirty four minutes. Another wonderful strike, an unstoppable low drive by Jack Barlow, was closely followed by a goal by Marc Cooper. Cleethorpes' number nine could easily have been penalised for a robust challenge on Newbould before bundling the ball home. The Garforth keeper never recovered and was withdrawn at half time.
Forward Luke Mangham emerged gloved up after the break and was quickly called into action with a great reaction save after Cleethorpes hit a post.
The teams played out a goal-less second half despite plenty of effort and Mangham was called on in the dying minutes to make another fine reaction save.