The Red Devils looked to have blown Thursday's make-or-break quarter-final second leg at Old Trafford, where the visitors responded to Manuel Ugarte and Diogo Dalot's first-half efforts by scoring four without reply.
Corentin Tolisso and Nicolas Tagliafico goals in quick succession took the match to extra-time, after Lyon kicked on despite Tolisso's sending off for two bookable offences as Rayan Cherki and Alexandre Lacazette rocked United.
But Amorim's men dug deep and went on to secure a scarcely believable victory as Bruno Fernandes's penalty was followed by Kobbie Mainoo's excellent 120th-minute leveller and Harry Maguire's match-winning header moments later.
"The sound of the two last goals were amazing and that is something we can keep for the future," the United boss said after an unforgettable 7-6 aggregate triumph that secured a semi-final against Athletic Bilbao.
"In the end you go to the semi-finals but you need to win it. If you go to the semi-finals and don't win it, it's the same feeling. We just need to focus on the next stage and try to win the competition.
"I think that (ending is) why we like this sport so much and all the frustration that the coach has in this kind of season, the frustration, the bad moments, when you had a moment like this it's all worth it.
"But then you finish, you calm down and then you start thinking we have a game on Sunday (against Wolves). Then we have to see all the players if they are fit to play because we have to focus on the Europa League."
The competition offers United's only route to silverware and European qualification given the side sit a lowly 14th in a campaign that is guaranteed to be their worst of the Premier League era.
Asked how big this moment can be, Amorim said: "I think these kind of moments can help a lot players in this kind of season, in this kind of context.
"They can create some connection between the fans and the players, and we can forget for a few minutes what kind of season we are having.
"Today is a good feeling but tomorrow we need to start everything again. So we need to focus on the next game, to focus on the players, if they are fit to play.
"We have to take the risk in to save some players because if you look at the our season, the most important thing is clear for everybody. It's the Europa League."
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