The Glovers struck a dramatic late winner to move a step closer to their first promotion to the second tier of English football.
Ed Upson nodded home with just five minutes remaining to complete the turnaround and sentence the Blades to a third successive season in League One.
Gary Johnson's men took just six minutes to wipe out the Blades' 1-0 first-leg lead when Upson's top-draw through ball teed up Kevin Dawson.
At the other end Jamie Murphy came within a lick of point of an equaliser. Callum McFadzean drifted through a posse of green shirts before teeing up the winger who cut inside before bending in a curling effort that thundered off the crossbar.
Jamie McAllister, a play-off finalist with Bristol City under Johnson's tutelage back in 2007, saw his drilled effort from a free-kick routine deflected wide after Harry Maguire had hauled back Paddy Madden.
Chris Porter, United's hero at this stage of last season's play-offs, fired into the side netting with the first chance of the second half after Dan Burn had done superbly to block his first effort.
Five minutes later Madden missed a gilt-edged chance from close range with the goal gaping after being found by Luke Ayling's delivery.
Maguire's free header with 20 minutes remaining landed well wide as the Blades searched for just their second goal in five games and a return to Wembley for the second successive season.
Yeovil have been late goal specialists this campaign with 20 scored in the final 15 minutes of games, but with time running out Hayter nodded wide when he rose to meet Foley's left-wing cross.
Midfielder Upson may have only scored one goal from inside the box for the club but he showed Hayter how it is done with just six minutes remaining. The former Ipswich man met Madden's inviting right-wing cross with a lovely glancing header which dropped into the bottom corner to send the Yeovil faithful wild.
With a minute remaining Foley's snapshot was just the wrong side of the post and the home side saw out the final minutes to extend Sheffield United's play-off hoodoo to seven attempts and tee up a second Wembley trip in five seasons for the ecstatic Glovers.