Rashford scored in extra-time, though had a chance to win the quarterfinal in the final minutes of normal time.
He said: "Listen to the noise, you can tell what a big game it is, a huge occasion. I haven't had the season I wanted but night likes this at Old Trafford are what it's all about.
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"I had to score that chance at the end, I wish it was done half an hour ago but we got the job done and we have to get something out of this season.
"Today represents what it always has to me, it's a massive game in the history of Manchester United. It gets us through to a semi-final but even a normal game against Liverpool we have to win.
"(Matchwinner) Amad Diallo has been working hard, he hasn't had the opportunities he wants and there's no better way to push for more than nights like this."