Like Shaw made the most of a rare start for Manchester United in finishing well from a Juan Mata through ball.
Paul Pogba opened the scoring with a third minute penalty for the Red Devils and Jamie Vardy headed home a consolation for Leicester.
And Shaw said scoring was the best moment of his career so far.
He said: “For me it’s the first one and that’s the most important. I’m up and running now
For my career so far, it’s probably the best feeling I’ve had.
“I want more feelings like that.
“That’s why you come to stadiums like this, to teams like this, to help the team and score goals like that.
“It was the best feeling today.”
Luke Shaw hopes this is the campaign he can finally make the left-back role his own.
On Shaw’s performance Mourinho said: “I am very pleased for Luke Shaw.
“He made one mistake in 90 minutes, he was very good defensively.
He had a good pre-season.”