Clarke gave the visitors a 14th-minute lead and struck again three minutes from time to give the visitors all three points after Dave Kitson had equalised in the 72nd minute.
Coventry's Carl Baker wasted a great early chance, firing off-target from a good position after Stephen Elliott's ball found him unmarked at the far post.
Elliott was the provider again when Clarke scored with a shot from around the six-yard box.
As Coventry continued to dominate, Baker had a shot charged down and Clarke fired in a low shot which was deflected wide.
Jonathan Forte, who has rejoined the Blades on loan from Southampton, shot straight at keeper Joe Murphy soon after coming on as a substitute early in the second half.
Kitson found the net with a header from Barry Robson's corner but Clarke struck the winner, seizing on the loose ball after George Long parried Franck Moussa's low drive.