United took the lead on the night and in the tie when Henrikh Mkhitaryan drilled in a low finish in the 10th minute but Anderlecht equalised when Sofiane Hanni scored after 32 minutes.
United were minutes away from a penalty shootout before Rashford got the winning goal after 107 minutes with a brilliant turn and finish.
That means Mourinho's men won 3-2 overall across both legs of the quarter-final.
Winning the Europa League is very important for United and manager Jose Mourinho as it offers another route into the Champions League, European club football's biggest competition.
The other sides into the semi-finals are Ajax, Lyon and Celta Vigo with the draw taking place at midday on Friday.

At a post-match conference the French Open champion was asked whether she knew which stars had been watching her play from the Royal Box at Centre Court.
David Beckham was a "great guy" she said.  But what about the other sportsman at the scene, Indian cricket legend Sachin Tendulkar?
She did not know who he was.
This is how Indians responded. #WhoisMariaSharapova is now trending worldwide.
As Sachin Tendulkar is clearly a god, then there is one perfectly understandable reason why Maria Sharapova would not know who he is - she must be an atheist, many tweeted.
Doctored images of Sharapova, picturing her hunched on the ground in apparent supplication in front of Tendulkar, made their rounds on the internet
One user @prasadvanjale urged God to "forgive her, for she doesn't know, what she is doing"  â€
Marcus Rashford's extra-time goal sent Manchester United into the Europa League semi-final beating Anderlecht 2-1 on a nervy night at Old Trafford.
[NEXT_CONCEPT]
Maria Sharapova may have been knocked out of Wimbledon, but in India she has hit the headlines with comments that are, for many Indians, akin to blasphemy.