Goals by Zlatan Ibrahimovic, Henrikh Mkhitaryan and Marcus Rashford eased United to a vital three points after drawing six of their previous nine league games.
United took the lead at a sun-drenched Stadium of Light when Ibrahimovic, receiving the ball with his back to goal, shook off his marker and buried a low 20-metre shot past Jordan Pickford.
Romero made a fine block to prevent Victor Anichebe equalising but matters went from bad to worse for Sunderland two minutes before the interval, with Larsson shown a straight red card for a lunging challenge that caught Ander Herrera on the shin.
Jose Mourinho made five changes for his side's 50th competitive game of the season but while keeper De Gea's absence was down to injury, the United boss opted to rotate players ahead of Thursday's Europa League quarter-final first leg with Anderlecht.
Ibrahimovic told Sky Sports: "Today was a good win and we have to keep focus, and keep working and keep believing we can reach the top four".
Sunday's victory left United only four points short of the Champions League places with eight games to go and a game in hand over Manchester City in fourth. "Afterwards we play with one more, sometimes it can be more hard and sometimes it can be an advantage".
With the result, United overtake Arsenal to fifth place on 57 points. There is a lot to be said as a football player for being ignorant to what is going on around you.
They look certainties for the drop to the Championship with just seven games remaining and having not found the back of the net for seven games. Whoever we speak about will be in the headlines because he plays in Manchester United and obviously everybody wants to play.
Sunderland vs Manchester United live streaming: The Sunderland is spotted at the last position in the Current Premier League table. "You could feel that negative feeling that is around a team that is close to relegation".
Koeman said he was pleased with his team's performance particularly the displays of Lukaku and Barkley, who had come in for some criticism after the Merseyside derby.
"We were hanging in the game and trying to do our best".
"It was good for him to play one hour with a good solid performance and no mistakes, so I'm really pleased for him", Mourinho commented after the easy win.
"These are the players available at the moment and, when I look to the others, I just expect Valencia to be ready to play and no more, I don't think".