Marcus Rashford scored a 94th minute penalty after a controversial VAR decision as United came from behind at PSG.
The win was the first time in the history of the European Cup that a team has overturned a 2-0 home defeat over two legs.
United got off to a dream start after just a minute when Romelu Lukaku pounced on a defensive error and rounded Gianluigi Buffon to give them a 1-0 lead.
But the celebrations were cut short just 10 minutes later as Kylian Mbappe got in behind the makeshift defence and squared for Juan Bernat to tap in an equaliser.
However, Rashford’s shot from 35 yards was spilled by Buffon into Lukaku’s path, and the striker tapped home United’s second after half-an-hour.
Despite only having 27 per-cent possession throughout, United kept pushing and earned a penalty in the 90th minute after Presnel Kimpembe handled Diogo Dalot’s shot with his arm.
Rashford slammed the penalty home and earned United’s spot in the quarter finals on away goals.
United’s unlikely progression has strengthened calls for Ole Gunnar Solskjaer to be given the job on a permanent basis.
He is currently acting as caretaker boss until the end of the season.
When is the draw?
The draw for the quarter-finals, semi-finals and final will take place on Friday March 15.
Teams will then know which side of the draw they are on.
It will take place at 11am UK time in Switzerland.
Which teams will be in the quarter-finals?
- Manchester United
- Bayern Munich/Liverpool
- Atletico Madrid/Juventus