The Red Devils made the breakthrough in the first half when Rashford managed to get on the end of Paul Pogba’s incisive pass before crossing for Henrikh Mkhitaryan.
The summer signing made no mistake with his finish after the midfielder fired a low shot into the net to give Manchester United a 2-1 aggregate lead.
However, Anderlecht were level before half-time thanks to Sofiane Hanni’s equaliser after confusion in the Manchester United defence.
Jose Mourinho’s side wasted a host of opportunities in the second half to restore their lead but the quarter-final tie went into extra-time at Old Trafford.
And Rashford scored a 107th-minute winner thanks to a superb solo goal which secured Manchester United’s place in the semi-final draw on Friday.
Speaking to BT Sport about Rasford’s performance after Manchester United’s close-fought win, Scholes said: “He was the one person you felt was going to decide the game and he did.
“His pace was great and he had good movement. He was really good.”
Scholes added: “You have to give Anderlecht credit. They put up a good fight.”
Manchester United are in fifth place in the Premier League table.
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