Captain Wayne Rooney, midfielder Paul Pogba and striker Marcus Rashford all took their shirts off and went over to the away fans at the final whistle.
“I told them to do that. It is Christmas time,” said Mourinho.
Two goals from Zlatan Ibrahimovic gave the Red Devils a 2-0 Premier League win against Albion at The Hawthorns.
It took the Swede’s tally to 11 for the season in the Premier League, with only Dutchman Ruud van Nistelrooy (12) scoring more in his first 16 league games for the club.
United are unbeaten in their past 10 games in all competitions and next face Sunderland on Boxing Day.
Mourinho added: “A shirt for a fan coming directly from the game with sweat means a lot.
“It is a pity that not everyone can get one but for the ones that can get it, it is a great feeling.
“For them [the players], it is just one more shirt. They will arrive in the dressing room and give it to the kitman, to a friend or change with a West Brom friend.”
Meanwhile, Watford sent a Christmas card to five-year-old Sunderland supporter Bradley Lowery, who is terminally ill. After leading out his heroes at the Stadium of Light against Chelsea on Wednesday night, he has subsequently received 28,000 Christmas cards.