Rashford struck the winner in extra time to send United through 3-2 on aggregate following Thursday night's 2-1 second-leg win, while Shaw impressed in his 120-minute performance at Old Trafford.
The Red Devils boss said that he was delighted with Shaw playing through the pain barrier, and also claimed that Rashford's attitude is the reason he has been played so much despite a dry spell in front of goal before this month.
"I always trust the players with special mentality," Mourinho told Sky Sports News. "Always trust them. Luke Shaw did something new today, it means a lot for me.
"After 90 minutes, before I know that I have to change Zlatan [Ibrahimovic], he had cramp, I had Ashley Young ready to come on, and he told me I'm going to play for 30 minutes with cramps, no problem. He said if you need to make a change in another position I am ready for it.
"These kind of things make me trust the players. There are technical aspects, but the mentality is the reason why I kept Marcus on the team all the time, because of that.
"At the performance level, he was months without scoring, but my trust on the boy is always based on his mentality. We have great examples of fighters, with the right mentality."
Man United will learn their semi-final opponent in Friday's Europa League draw at noon BST, with Celta Vigo, Lyon and Ajax joining the Premier League outfit in the pot.