Shaw, often criticised this season, played 120 minutes as United edged into the Europa League semi-finals with a 2-1 extra-time win at Old Trafford on Thursday.
Marcus Rashford was the hero with his 107th-minute goal, but the win came at a price for the hosts, who lost Zlatan Ibrahimovic and Marcos Rojo to injuries.
Mourinho was full of praise for Rashford – who now has 10 goals in all competitions this season – and Shaw.
"I always trust the players with special mentality. Always trust them. Luke Shaw did something new today, it means a lot for me," he told a news conference.
"After 90 minutes, before I know that I have to change Zlatan, 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."
Rashford had been goalless in the Premier League since September before scoring in back-to-back games earlier this month.
His match-winner sent United into the last four and Mourinho urged his players to retain belief in their ability to go on and win the Europa League.
"It was a difficult game, a difficult opponent, I am tired, I imagine the players are more tired than me," he said.
"But we are in the draw [on Friday] and we still have the dream to win the competition."