Victory saw United close to within four points of Manchester City, who occupy the fourth and final Champions League qualifying berth and have played a game more than Mourinho's side.
Zlatan Ibrahimovic, Henrikh Mkitaryan and Marcus Rashford all scored as the Red Devils eased past struggling Sunderland.
They're now in fifth place, four points off the pace of the top four behind Manchester City, a club with which they have a game in hand. "When you play a team that is sad, if you score first the game is nearly over because it is hard for them to react".
Winning the tournament earns them a spot in next season's Champions League, so it's just as good as a top four Premier League finish.
Mourinho has been a vocal critic of the 21-year-old's in recent weeks, suggesting at one stage that he was the full-back lowest down his pecking order.
Mourinho embraced Shaw and appeared to give him words of encouragement after replacing him with Daley Blind just past the hour mark, and the Portuguese said afterwards that the ex-Southampton man played well.
"He played well, played solid. For him to get 90 minutes today was good after being out for such a long time".
"Every single game is massive now, but I think the one on Thursday is huge".
"It is wonderful to have a great player like him in our team and hopefully he keeps going until the end of the season and helping us".
Manchester United's Zlatan Ibrahimovic, left, and Sunderland's Jason Denayer battle for the ball during their English Premier League soccer match at the Stadium of Light, Sunderland, England, Sunday, April 9, 2017.
"He has a problem with his hip", Mourinho told BBC Sport.
Mourinho left David De Gea out of the squad for the Sunderland game but said he was confident he would be fit in time for next Sunday's meeting with Chelsea, with Sergio Romero set to continue in goal against Anderlecht.
Well, there is little doubt that de Gea has improved drastically since when he first moved to United back in 2011.
Sunderland remains bottom having now failed to score in its last seven games, 10 points adrift of safety. We want to fight for trophies.
The former Chelsea boss hopes to have as many players as fresh as possible ahead of that trip to Belgium as United look to secure a commanding first-leg advantage.
But man-of-the-match Marcos Alonso made the game safe with an exquisite free-kick and Conte said: "Yes, I'm satisfied because it wasn't easy to play this game after the win by Tottenham but our answer was very good".