AVA- Never in Europe's premium club competition had a team progressed after a 2-0 home defeat but Rashford's stoppage-time penalty made sure Ole Gunnar Solskjaer's side advanced in the most dramatic fashion.
The visitors, severely depleted by injuries, went ahead twice in the first half through Romelu Lukaku but Juan Bernat equalised for the French champions in between.
PSG were advancing to the last eight 3-2 on aggregate before referee Damir Skomina awarded United a penalty for a Presnel Kimpembe handball after a lengthy delay for the VAR review and Rashford lashed home the spot-kick to level the tie at 3-3 and send United through on away goals.