The Premier League side went into the clash as heavy underdogs after falling to a 2-0 defeat at Old Trafford, but an early goal from Romelu Lukaku gave United hope, the Belgian finding the net after just two minutes.
PSG enjoyed plenty of possession down the left flank in the early stages, however, with Eric Bailly and Ashley Young's struggles seeing Juan Bernat find an equaliser just 10 minutes after Lukaku's opener.
A second first-half goal from the striker restored the faith, but it would take a contentious penalty in the 94th minute, confidently converted by Marcus Rashford, to clinch progression to the quarter-finals via the away-goals rule in a 3-3 aggregate scoreline.
In securing progression, United became the first side in Champions League history to recover from a 2-0 first-leg defeat at home, and the dressing room at Parc des Princes was understandably jubilant in the wake of the victory.
The Red Devils' celebrations hold extra weight in that a mass of first-team players including Juan Mata, Ander Herrera, Jesse Lingard, Anthony Martial and Nemanja Matic were all forced to miss the crunch clash through injury.
As such, calls for interim boss Ole Gunnar Solskjaer to be handed the reigns full-time have hit an all-time high since the Norwegian took over from Jose Mourinho, with the Old Trafford legend already having broken several records during his time in the dugout.
But while fans have one eye on United's opponents in the last eight – which currently include Tottenham, Ajax and Porto – Solskjaer must steer his squad's attention back to the Premier League, with a trip to Arsenal on Sunday posing another test of the club's mettle.
Occupying fourth spot in the English top flight, United sit one point ahead of the Gunners and a victory would go some way to stamping their name on a spot in the Champions League again next season.
But while the Premier League title may be out of reach, Solskjaer and his charges are in with a chance of domestic silverware in the form of the FA Cup, as Wolves lie in wait for a quarter-final clash on March 16.