The Blades went 1-0 and then 2-1 down in the first half before a stunning second-half fightback saw them close the gap between themselves and leaders Burnley, with Oli McBurnie on the scoresheet twice.
Jack Robinson also scored, to redeem himself after his error led to Burnley’s second, with Anel Ahmedhodzic and Iliman Ndiaye also on target.
Here’s how we rated the Blades’ players on a five-star afternoon at Bramall Lane...
9. Ben Osborn 7
Returning to a more natural midfield position after a spell as an emergency wing-back and did superbly well at full-stretch to deny Cullen a good sight of goal just after United had dragged themselves level for the second time Photo: Bradley Collyer
10. Iliman Ndiaye 8
In the right place at the right time once again to equalise for the first time, in a Billy Sharp-type position to finish past Muric with his knee or thigh - whatever it was, he or United didn't care. Thought he had a second in the second half but he was flagged offside - correctly - after Muric saved McBurnie's header Photo: Andrew Yates
11. Oli McBurnie 9
Came back into the side as one of three changes made by Paul Heckingbottom and justified it with two goals and a superb individual display. Deservedly given a standing ovation as he came off after scoring his eighth and ninth of the season. Photo: Andrew Yates
12. Subs: Chris Basham
Replaced Stevens and operated in midfield rather than the backline Photo: Ashley Crowden