Amid large fan protests ahead of the game, Chelsea were in desperate need of a win as they chase a top-five finish.
Matheus Cunha’s first-half goal was all Manchester United needed to secure all three points and put more pressure on Rosenior.
Chelsea have now lost four games in a row without scoring a goal for the first time since 1998.
"It is so difficult. Today, they had one shot on target while we were down to 10 men," Rosenior reflected.
“We had wave after wave of attack and hit the woodwork, I think four times. I don't want my team to feel things are against us. We have to keep fighting.
"We have to defend that moment better. We don't, and we get punished. At the moment, any small mistake we make and the ball ends up in the back of our net, and that has to change.
"It is about results, and we haven't got a result today we should have. The small things add up, but we have to keep working hard.
“I put myself under the most pressure. We just have to keep working with the staff and the players to turn things around."