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United had been cruising at 2-0 up with 20 minutes of the 90 to play, only for an astonishing collapse to see it taken to extra-time and go 4-2 down by the 109th minute.

However, Bruno Fernandes's penalty restored some hope, then Kobbie Mainoo's 120th-minute curler and Maguire's back-post header a minute later sealed the most remarkable of European comebacks.

It means Ruben Amorim's hopes of winning silverware in his first season at the helm remain intact, with a semi-final against tournament favourite Athletic Club to come.

United produced an excellent first-half performance and led within 10 minutes. Alejandro Garnacho stepped on to Fernandes's pass on the right side of the area and cut the ball back for Manuel Ugarte, who tapped into a gaping goal.

Only a full-stretch save from Andre Onana denied Rayan Cherki a glorious equaliser, then Fernandes rattled the crossbar with an over-the-shoulder volley at the other end.

But United got the second goal their first-half pressure warranted in stoppage-time, as Diogo Dalot outmuscled Nicolas Tagliafico and finished low across goal.

Lyon introduced some nerves through Corentin Tolisso's close-range header with 19 minutes of the 90 remaining, though, then Tagliafico's shot crept over the line despite the best efforts of Onana six minutes later.

Despite Tolisso receiving his second booking just before the 90 minutes were up, Lyon continued to press in extra-time and went ahead through Cherki's left-footed drive in the 104th minute.

A penalty conceded by Luke Shaw and converted by Alexandre Lacazette had Lyon seemingly coasting into the semi-finals at 4-2, but when Thiago Almada kicked the bottom of Casemiro's boot, Fernandes kept his cool from 12 yards to offer United a lifeline.

From there, the Red Devils staged the most remarkable of rescue acts, Mainoo curling home to make it 6-6 on aggregate before Maguire nodded Casemiro's hanging cross home to spark wild celebrations at Old Trafford.

Amorim said watching a documentary about Manchester United's 1999 UEFA Champions League win provided the inspiration for the miraculous fightback.

United have history when it comes to European comebacks, memorably beating Bayern Munich 2-1 having trailed 1-0 in second-half stoppage-time in the 1999 UEFA Champions League final, but it produced arguably its best rescue act of all against Lyon.

"It was a great night. The team was tired and you could feel it in the game, at 4-2, even with one more player, you felt it was over. But here, it's never over, Amorim said.

"Here, everything is possible. You feel the environment, then after the penalty from Bruno, you felt we could change the game.

"We put Harry Maguire up there because he's the only guy that can score a goal with his head.

"Then Kobbie Mainoo lacks some pace at the moment, but we put him there because he's very good in short spaces. We changed things and today it worked.

"We started well but we were not consistent enough to maintain it. When the team is tired, we drop a lot.

"We have seen against Arsenal and Manchester City that we can defend well. We should do better with the first two goals of Lyon and we have a lot to improve, but the character was there." 

With several players, including Luke Shaw and Mason Mount, returning from injury, the United boss thinks things are looking up for his side.

"It's really important. You feel now that playing European and Premier League games is really hard," Amorim said of his returning players.

"If you don't have the full squad, it's almost impossible. We brought on Mason Mount, Luke Shaw was only supposed to play 30 minutes."