Romelu Lukaku broke the deadlock inside three minutes, rounding the goalkeeper after a defensive mistake.
Juan Bernat pulled one back for PSG but Lukaku was on hand to restore the lead as Gianluigi Buffon blundered.
With an aggregate scoreline of 3-3 from the two fixtures, United progress to the quarter finals on away goal.
Paris Saint Germain 1-3 Manchester United (3-3agg): Last-gasp Marcus Rashford penalty from VAR decision sees Red Devils progress to quarter-finals after dramatic showdown at Parc Des Princes
Manchester United sensationally progressed to the quarter-finals of the Champions League with a 3-1 win at PSG.
Romelu Lukaku's first-half double and a dramatic, last-gasp Marcus Rashford penalty sent them into the last eight.
But VAR will once again take the headlines with PSG furious at the late handball call.
Lukaku opened the scoring in just the second minute before Juan Bernat levelled for the Parisians.
The Belgian scored his second later in the first half before Rashford kept his cool from 12 yards on a famous, famous night for United.
Here's how the players rated in the Parc des Princes...
David de Gea
Made a routine stop from Juan Bernat at 1-1 but was helpless to prevent the Spaniard's equaliser on 12 minutes. 6
Ashley Young
A surprise selection at right wing, he barely protected the erratic Eric Bailly and eventually moved to right-back after Diogo Dalot came on. Improved there. 5
Chris Smalling
Made two crucial interventions shortly after United went ahead. Quickly regained form after a weekend wobble against Southampton. 8
Victor Lindelof
Was under the cosh for the first half-an-hour but led well alongside Chris Smalling as they relatively reined in Kylian Mbappe. 8
Eric Bailly
A walking disaster. Did not do a single thing right and should not have been played at right-back. Forced off injured after a Marco Verratti foul. 1
Luke Shaw
So lax at times in defence. Appeared intimidated by PSG's forwards and hardly expressed himself further forward. 6
Scott McTominay
Needed to play deeper to protect the back four and did not seem settled in a 4-4-2. Coped commendably, given he was up against the majestic Marco Verratti. 5
Andreas Pereira
Positioned on the left wing in the 4-4-2 and did not look comfortable there. Took a long time to find his stride and barely contributed in attack. 5
Fred
Possibly his most impressive performance yet for United. Operated at a quick tempo, pressed aggressively and linked up well. 8
Marcus Rashford
Had his shot fumbled by Gianluigi Buffon which led to Lukaku's second. Regularly threatened in the first-half but poor in the second. Then he scored. 7
Romelu Lukaku
Recovered from an imperfect first touch to superbly beat Thiago Silva and then Gianluigi Buffon for the first goal. Pounced lethally for the second. 9
Subs
Diogo Dalot
Should have started the game and almost created a goal before half-time. 6
Tahith Chong
Came on near the end. 6
PSG: Buffon 4, Kehrer 5 (Paredes 70, 5), Thiago Silva 6, Kimpembe 6, Bernat 7, Marquinhos 6, Verratti 7, Alves 6, Draxler 6 (Meunier 70, 5), Di Maria 6, Mbappe 7
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