Four days on from leaving Shaw out of his matchday 18 for the 0-0 draw at home to West Brom and questioning the 21-year-old's professionalism, a second-half injury to Ashley Young forced him into the fray with United 1-0 down to Phil Jagielka's 22nd minute opener.
The former Southampton youngster was in the thick of the action in stoppage time when his rasping strike was handled in the six-yard box by Ashley Williams, bringing a red card for the Everton centre-back before Zlatan Ibrahimovic converted the resulting penalty.
But, speaking to BT Sport after the match, Mourinho made it clear that Shaw still has plenty to do to win his favour.
"I think he has lots of potential but the football brain and the professional brain has to be in with the talent," he said, after the point left his team four points behind Manchester City in fourth.
"I think in previous generations it was more difficult to be a big player, more difficult to be rich - rich after the career, at this moment they are rich when the career starts. It is not an easy life for them.
"I think he has to change his football brain. I was telling him just now, he was doing things in the second half because he was reacting to my voice. If he was playing on the other side, for sure he would not do it because I was not there to think for him.
"He needs maturity and maturity comes with responsibility. He has talent, I want to help him. I think he had a future here.
"But Manchester United cannot wait. We are 20 matches unbeaten in the Premier League – it is not enough because we don't win enough matches."
Shaw was not the only Old Trafford youngster to feel the sharp end of Mourinho's tongue, as Marcus Rashford extended his run without a Premier League goal to 20 appearances.
"We need the kids to grow up," the former Real Madrid boss said.
"One thing is for Marcus Rashford to come in last year with no responsibility and score a goal, then the next week score another one.
"This season he scored the last one in September. I have to help him, I cannot kill him. I must help him, I must try to accelerate the process again because this is Manchester United.
"The good thing was the spirit of the second half, following a first half that was not good at all."
One man who has no trouble finding the net in United colours in Ibrahimovic, whose penalty was his 27th of the season in all competitions.
The 35-year-old had a header chalked off incorrectly for offside in the 71st minute and Mourinho felt this marginal call going the wrong way strengthened the argument for the introduction of video assistant referees (VARs) following FIFA's recent trials.
"They [the United players] fought until the last second and with the VAR we win this game 2-1 because it is not office. It's a difficult one for the linesman, I'm not criticising him at all," he added.
"When the VAR comes it will help everyone, especially them.
"From what I saw just now on the computer, for me it is not an offside. But I repeat, not at all am I criticising the linesman, not at all.
"It will be welcomed by them because it is not good when they make mistakes , even a little one like this."