PSG, meanwhile, find themselves dumped out of the competition in the knockout stage for the third consecutive year, reports Efe news.
Starting with a 2-0 advantage after their hard-fought victory at Old Trafford, PSG on Wednesday looked poised to break their Champions League curse and justify the fortune the club's Qatari owners have spent in pursuit of a European title.
And while injured PSG superstar Neymar had to watch the game from the stands at Parc des Princes, United were without Paul Pogba, Juan Mata, Ander Herrera and Alexis Sanchez.
With so many holes in his roster, visiting coach Ole Gunnar Solskjaer took a conservative approach in terms of formation and tactics.
But a colossal mistake by PSG defender Thilo Kehrer left goalkeeper Gianluigi Buffon at the mercy of Romelu Lukaku and the Belgian international scored to cut the French club's lead in half in the 2nd minute under driving rain in Paris.
Instead of pressing for more, United fell back and ceded possession to PSG for an extended spell. In the absence of Neymar, Dani Alves and Kylian Mbappe spearheaded the PSG attack and made life difficult for United's makeshift back line.
PSG restored the two-goal margin in the 12th minute, when Juan Bernat collected a cross from Mbappe and blew past Eric Bailly to beat David De Gea. Mbappe hit the post in the 16th minute, four minutes before De Gea denied Bernat, while Angel Di Maria's effort, in the 21st minute from a distance, went wide.
Failing to secure the tie, PSG conceded again at the half-hour mark. This time, Buffon had only himself to blame, as his failure to deal with a long strike by Marcus Rashford set up Lukaku's second goal of the night.
Though United led 2-1 on the night, PSG were still up 3-2 overall. De Gea bested Mbappe in the one-versus-one, and the post deflected a Bernat shot to keep United's hopes alive.
Four minutes into stoppage time, a shot by United's Diego Dalot bounced off the arm of defender Presnel Kimpembe out of play for a corner.
Yet after reviewing the play with the Video Assistant Referee, the official cited Kimpembe for hand ball and awarded a penalty.
Rashford stepped to the spot and converted to level the aggregate score at 3-3 and give United the edge on away goals.