Ole Gunnar Solskjaer's team, outplayed for most of the game 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 when 19-year-old Brandon Williams scored his first senior goal in the 72nd minute.
Substitute Mason Greenwood, 18, equalised minutes after coming on and with United now flying forward, Rashford swept home in the 79th minute.
But still 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.