The big centre-back snatched the winner with a textbook header from Phil Jagielka's deep cross after Nick Mongtomery had given the Blades the lead.
Stoke, stung by two successive defeats over the Christmas period against Burnley and Leeds, had United in trouble in the opening minutes.
Midfielder Luke Chadwick, on loan from West Ham, whipped in a cross from the right, Paul Gallagher's right-foot volley hit Sambegou Bangoura but the Guinea international recovered to finish with a close-range strike which was pushed around the post by Paddy Kenny.
Chadwick went close with a right-foot effort from 16 yards before Gallagher, on a season's loan from Blackburn, ran on to another Chadwick cross before slamming a half-volley over the target.
United swept ahead in the 18th minute after Gallagher fouled full-back Derek Geary on the right flank. Skipper Michael Duberry half cleared the free-kick to the lurking Montgomery and the United midfielder had time to pick his spot past the unsighted Steve Simonsen from 20 yards.
Darel Russell squandered an opportunity by side-footing high over the United bar after neat work from Chadwick before Montgomery was handed another chance only to drive high and wide from 16 yards after David Unsworth's corner caused havoc.
Stoke were back on level terms in the 57th minute after the United defence allowed Bangoura too much space. He threaded the ball to Chadwick whose shot deflected toward Mamady Sidibe who coolly rounded Kenny to slot the ball into an empty net for his fourth of the season.
Kenny's outstretched leg then kept out a Russell shot before Morgan had the last word.