The defender has often come in for heavy criticism from Mourinho and is tipped to leave Old Trafford in January, but the Reds boss was full of praise on this occasion.
Shaw was handed a chance in the Champions League clash and grabbed it with both hands as United progressed through to the last 16 of the competition as Group A winners.
"Luke Shaw, offensively, very, very good and great contribution for the team," he added. "Romero is an amazing goalkeeper and is only on the bench because we have David (De Gea).
"(Shaw) Was waiting for quite a long time, he was deserving for quite a long time, it is not easy to give an opportunity to a player who doesn't play for many months.
"Honestly, I'm happy with his performance."
Not even Alan Dzagoev's controversial opener had the Old Trafford faithful concerned as only a record seven-goal defeat would deny United a place in the last 16.
And quick-fire goals from Romelu Lukaku and Marcus Rashford after the break turned the contest around as United triumphed 2-1.
"We confirmed the way we wanted to confirm, which was to play well and have a good result," Mourinho told BT Sport 2.
"Half-time we were losing but playing well, I was not unhappy with the players at all. In the second half we had that strong reaction, we didn't want to lose the match.
"The boys have played professionally and qualified the way we wanted to qualify."