The Welsh side were kept at bay by United keeper Paddy Kenny, who was playing his first game of the season after serving a nine-month suspension for testing positive for a banned substance.
He pulled off a string of saves before the Blades side scored twice after the break through Richard Cresswell and an own goal from the unfortunate Dorus de Vries and with Blackpool winning at Peterborough, Swansea must beat Doncaster to keep their hopes of a play-off berth alive.
Swansea, the Championship's lowest scoring side, almost snatched the lead on 26 minutes.
Mark Gower drilled the ball across the face of the United goal but Shefki Kuqi and Cedric Van Der Gun both failed to add the finishing touch.
United were in trouble again when the lively Nathan Dyer pumped the ball across the field to the unmarked Gower whose right-footed strike was scooped up by Kenny.
Swansea then had claims for a penalty waved away referee Scott Mathieson minutes before the break.
Kenny's remarkable comeback continued as he conjured up a spectacular stop to tip Federico Bessone's curling free-kick around the post with the deftest of touches.
Kenny hampered Swansea's hopes of a breakthrough with another vital save, leaping at the near post to tip away Bessone's teasing free-kick under pressure from Kuqi before the United keeper pulled off another stunning save stretching to snatch Pratley's long-range strike.
Cresswell struck in the 64th minute after Ashley Williams' blunder was gratefully accepted by Glen Little.
He then teed up substitute Mark Yeates who provided the vital cross from the left flank and the former Stoke striker cooly controlled the ball before hammering it into the roof of the net.
De Vries was then left helpless as Cresswell wriggled his way into the area in the 89th minute before crossing only for the keeper to divert the ball into his net.