Liya Rafeeq has also won a full scholarship from her dream university, New York University Abu Dhabi (NYUAD)
Ole Gunnar Solskjaer's team, outplayed for 70 minutes by the promoted side, trailed 2-0 after a first-half goal from John Fleck and a second-half effort from Lys Mousset.
Chris Wilder's team had been hungrier and more incisive all over the pitch but, from nowhere, United were back in the game and when 19-year-old Brandon Williams scored in the 72nd minute.
Manchester United suddenly looked full of ideas, with the game turned on its head.
Solskjaer threw on Mason Greenwood to replace Andreas Pereira, meaning United were operating with an attacking quartet of Greenwood, Marcus Rashford, Anthony Martial and Daniel James.
Just five minutes after Williams' strike, Greenwood equalised from close range after Rashford's cross.
With the home side rocking badly, United poured forward again. Rashford swept home in the 79th minute.
The drama was not over, with Oliver McBurnie controlling the ball and firing past David de Gea to level at 3-3 in the 90th minute. The goal stood after a VAR check for handball.
The result means Manchester United are ninth in the Premier League table, nine points behind fourth-placed Chelsea while Sheffield United are sixth, a point ahead.
Earlier, Solskjaer's men were indebted to goalkeeper De Gea, who produced a stunning double-save to keep the score at 0-0 early in the game.
First he got down low to his right to keep out a shot from John Lundstram before recovering seconds later to parry David McGoldrick's point-blank header.
Solskjaer's team looked rattled, with Williams, on just his second Premier League start, making a couple of early errors and Phil Jones, on his first league appearance of the season, looking like an accident waiting to happen.
Jones was at fault for the opener. A long ball down the right wing put him under pressure but he should have seen off Mousset.
Instead, the Frenchman outmuscled Jones and cut the ball back for Lundstram, who saw his effort well saved by De Gea but the goalkeeper could do nothing to keep out John Fleck's follow-up.
Fred and Pereira were unable to establish any control in midfield. United's forwards were barely able to get into the game, mustering just one shot in the whole of the first half.
Solskjaer replaced Jones with Jesse Lingard at half-time but Sheffield United deservedly doubled their lead in the 52nd minute.
Manchester United lost the ball on the left and Fleck delivered an inch-perfect pass to Mousset, who bent a right-footed shot past the diving De Gea from outside the box.
To chants of "There's only one United," Sheffield continued to boss the match against their insipid opponents, who looked short of confidence.
With the game drifting, Manchester United found a foothold when a James ball into the box fell to Williams, who lashed home on the half-volley.
Greenwood then levelled and Rashford gave the visitors an unlikely lead in a game in which they had largely been outplayed.
But just when it appeared Manchester United had grabbed a morale-boosting win, McBurnie struck.
Liya Rafeeq has also won a full scholarship from her dream university, New York University Abu Dhabi (NYUAD)
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