Clarke shot unchallenged from 25 yards past goalkeeper Paddy Kenny and into the bottom right-hand corner to secure Stoke's first victory over the Blades in 14 attempts.
United left the field to a chorus of boos as the fans voiced their displeasure after a poor display belied their promotion credentials.
United should have gone ahead in the eighth minute when Andy Gray, looking suspiciously offside, screwed his shot wide after finding himself one-on-one with the goalkeeper after Peter Ndlovu forced the ball through.
Carl Asaba could have opened the scoring on his first return to Bramall Lane after joining Stoke in the summer. But the striker pulled the ball inches wide after being played in by Gifton Noel-Williams in the 11th minute.
A rare moment of quality saw United's top scorer Jack Lester bamboozle two defenders before unleashing a shot which flashed across the face of goal, and a minute later he curled a shot inches wide.
Kenny then punched out a fierce Kris Commons shot from the left of the area as the game threatened to burst into life midway through the half.
Wayne Thomas clashed with Lester off the ball on the edge of the area. A shoving match ensued as Lester stayed down but no action was taken, sending United boss Neil Warnock into an arm-waving frenzy on the touchline.
Clarke sent a 30-yard shot over before giving Stoke an unexpected lead to silence the home fans.
Early in the second half United substitute Jonathan Forte's shot fizzed along the turf from 10 yards after a scramble in the area following Michael Tonge's free-kick but goalkeeper Neil Cutler smothered the shot.
Robert Page almost put through his own net after the visitors broke, taking advantage of the gaps that were appearing as United threw everything forward.
Lester had the ball in the net after he turned and side-footed home from close range but that was disallowed for a push on Stoke defender Gerry Taggart.
Andy Gray set off on a 40-yard run but with no support he was forced wide and his tame shot was easily held by Cutler.
United then had a let-off when Noel-Williams blazed over from close range and the visitors were showing more composure and incision.
The Blades appeared bereft of imagination and the game petered out with the forlorn home players leaving the field to a chorus of disapproval.