The Premier League club entered the night two goals down from the first leg, but a controversial VAR decision and Rashford's subsequent penalty strike handed them a 3-1 win on the night and sent them through on away goals following a 3-3 aggregate scoreline over the two matches.
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And while the 21-year-old had never taken a penalty in a competitive match for the club, and had the win riding on his shoulders late on, he claims moment's like Tuesday's spot-kick are what players "live for".
"I think it was just about keeping a cool head," he told BT Sport . "That's what you practice for. I wanted to take it.
"Those moments are the moments we live for, and we want to come out of them smiling after.
"Everything seemed to be against us, but we're used to surviving these moments and we proved that again. We keep pushing and bettering ourselves.
"It's a beautiful thing, to have 10 players injured too, but it’s an opportunity and it's what they deserve."
Rashford wasn't alone on the night of course, strike partner Romelu Lukaku having already scored twice to set up the late winner for the youngster.
Lukaku's second came from a spilled Rashford strike, and following the match Lukaku said it was all part of a belief between the two forwards coming into the night.
"Me and Marcus Rashford knew we could do the business today after playing together against Fulham," the Belgian told BT Sport . "It was a pleasure to play with him today like that.
"We agreed to step up the press together and I told him every time he shoots I would follow the ball in.
"We just keep going - everything starts on the training ground with 5 v 5, 6 v 6 and the competitiveness starts there. It's natural to us, we keep going until the referee blows the whistle."
Manchester United join Tottenham, Ajax and Porto as clubs to have booked their place in the quarter-finals, with the other four slots to be decided next week.
But before then the Old Trafford club will have to focus back in on the league, where they face a showdown against fellow top-four chasers Arsenal on Sunday.
