The Red Devils, without 10 first-team stars including Paul Pogba and Jesse Lingard, came from 2-0 down from the first leg to reach the Champions League quarter-finals.
Romeu Lukaku scored a first-half brace either side of a Juan Bernat goal, but a dramatic injury-time sequence saw United awarded a penalty for a handball on Presnel Kimpembe.
Marcus Rashford stepped up to fire past Gianluigi Buffon and make it 3-1 on the night and 3-3 on aggregate and complete a remarkable turnaround to put United into the last eight.
That sparked jubilant scenes amid the players and away fans and among those celebrating in the stands were several United icons, including Cantona, Patrice Evra and Sir Alex Ferguson.
And Cantona told United’s official Twitter account post-match: “I love it. I’m so happy! I’m so happy.”
Ole Gunnar Solskjaer has now won 14 of his 17 matches in charge and is seen as a shoe-in to take up the job on a permanent basis in the summer.
The Norwegian boss has steered United into their first quarter-final since 2013-14 and despite being written off pre-match, he insists they always had faith they could overturn the first-leg deficit.
“We always believed. That’s the thing,“ Solskjaer told his post-match press conference.
“That’s the thing. We set out the plan, it wasn’t about having the ball and outplaying them.
“Because if you give them too much space and time, with [Kylian] Mbappe, you saw a few times in the first-half when we missed our defensive shape they threw us straight away.
“The plan was to get the first goal, be in the game with five or 10 minutes to go and we were.
“The belief in the boys was just what we hoped for. Of course, there was a great start. We set out a plan to get the first goal.
“We didn’t expect it after two minutes, but that made it more open for us. We wanted many goals in the game.”
United last won the European Cup in 2009 and this win even has fans dreaming of another triumph on the continent.
Asked about their chances of a Champions League win, Solskjaer said: ”Of course we fancy ourselves.
“We can go all the way but we just have to wait for the draw and then take the game as it comes.
“We’ve had loads of injuries, suspensions, but that was maybe a good thing today.
“We had a fresh team, loads of energy, and we defend and humble enough to work hard and we had a team to do that.”