Former Blades loanee Clarke tapped in with 13 minutes remaining of Tuesday night's League One clash following an error by United defender Neill Collins before substitute Hayes seized on Harry Maguire's hesitation to seal it.
The home side twice hit the woodwork but failed to score at home for a fourth successive game.
Winger Ryan Flynn almost broke the deadlock four minutes before the interval but his low angled drive from the right rebounded off Paul Jones' far post.
Crawley forward Clarke was denied on the stroke of half-time when his close-range header was instinctively repelled by George Long.
Blades captain Michael Doyle rattled the Crawley woodwork for a second time minutes after the restart as his dipping left-foot volley thumped off the right-hand upright.
But the visitors went on to win it as striker Hayes robbed Collins midway inside the United half to set up Clarke before sliding in the second after dispossessing Maguire on the edge of the box.