Just 22 days after suffering his only defeat as caretaker manager in the first leg, United arrived in France looking to become the first side in the competition’s history to progress after suffering a first-leg home loss by two goals.
The odds were stacked against them – but United turned the improbable into reality in an extraordinary Parc des Princes clash, securing a 3-1 victory to progress on away goals.
Romelu Lukaku scored either side of Juan Bernat’s effort in a first half which had Solskjaer’s side dreaming, with the video assistant referee’s stoppage-time time intervention leading to a penalty that Rashford slammed home.
United were without 10 players at the big-spending Ligue 1 champions, but those selected embodied the positivity displayed by their manager and spirit synonymous with the club over the years. The visitors flew out of the blocks and Lukaku capitalised on a mistake within two minutes, slotting home after showing strength and skill.
Thomas Tuchel’s side rallied and looked set to roar ahead after Bernat turned home Kylian Mbappe’s cross, only for Gianluigi Buffon to spill a hopeful Rashford shot into the path of Lukaku.
It was a frantic first half followed by a far edgier second period and United’s hopes of reaching a first quarter-final since 2014 looked to be over - until added time.
Presnel Kimpembe was adjudged to have handled a Diogo Dalot shot after referee Damir Skomina looked at the incident on a monitor – and Rashford did the rest.
The celebrations at full-time reflected the magnitude of the win and could well prove to secure Solskjaer the job permanently.
Rashford, when asked what he was thinking as he stepped up to take the penalty, said: “Just keep a cool head. They are the type of thing you practice every day and I wanted to take it.
“That was probably the hardest thing, the wait before, but it’s all part of the game and those moments, that’s what we live for.
“Everything seemed to be against us but we’re used to surviving in these moments. Today just proved that one more time.
“We go forward from this game now, keep pushing and better ourselves.
“It’s a beautiful thing (seeing the club’s youngsters on the pitch). We have 10 players injured but for them, it’s an opportunity for them to come forward and they’ve all done that.”
Interim boss Solskjaer said: “Fantastic. The belief in the boys was just what we hoped for.
“It was a great start. We started with a plan to get the first goal, but we didn’t expect it to come so soon.
“I was in the Champions League with Molde and we got a penalty in the last minute and my player missed and we went out.
“Rashford, he’s 21, there were no nerves whatsoever, fearless.”