Darren Bent struck twice either side of a Darren Ambrose goal, to cancel out strikes from Peter Ndlovu and Dean Windass which had given the Blades a seemingly unassailable lead.
Bent had wasted an early one-on-one chance, shooting straight at Paddy Kenny, before strike partner Pablo Counago foolishly got himself sent off in the 20th minute.
After tangling with Robert Page, the Spaniard struck the defender across the back of the head to earn his second dismissal of the season.
Windass wasted a clear heading chance from a Michael Brown cross, before Ndlovu gave the visitors a half-time lead in stoppage time.
Phil Jagielka sent him dashing down the left flank, before he cut inside to fire past Andy Marshall.
Straight after the interval, Tommy Miller's ambitious lob bounced off the crossbar, before United went 2-0 up in the 50th minute.
Jermaine Wright conceded possession to Michael Tonge who sent Windass racing up to the edge of the box, from where he drove low beyond Marshall.
Town were given a lifeline in the 57th minute when Darren Bent volleyed home Wright's chip into the penalty area.
The goal lifted the hosts and they proceeded to peg United back for long spells of the second half.
They deservedly equalised in the 78th minute when Shaun Murphy's clearance was blocked by Marcus Bent, who chipped a cross for an unmarked Ambrose to head beyond Kenny.
The miraculous comeback was sealed with three minutes left when Herman Hreidarsson played in Ambrose who crossed to the far post where Darren Bent thundered home a header for his 13th of the season.
In stoppage time United almost grabbed an equaliser when Thomas Gaardsoe's header was volleyed just past the far post from 25 yards by Brown.