Moyes replaced Sir Alex Ferguson at Old Trafford this summer and United have claimed just 10 points from their opening seven Premier games under his tenure.
The Red Devils looked to be slipping to a fourth league defeat of the season on Saturday after a poor first half against Sunderland before two goals from Adnan Januzaj handed them a 2-1 win.
And while Rooney believes a period of transition is inevitable with a new manager. he still insists the players have to do better for their boss.
"Obviously the manager is under a bit of pressure, but we know as a team it's us who have let him down," he told national newspapers.
"We have to pull together for the manager and show the sort of fight we have done here and we'll be fine.
"This will be a transition season, because it's a new manager with different styles and we have to adapt to them.
"More importantly we need to do better for him than we have been."
Moyes handed Januzaj a surprise first start against the Black Cats and the young Belgian repaid his manager's faith with a stunning performance as well as his two goals.
Rooney admits that throwing the 18-year-old into a struggling team could have been considered a risk but was pleased that the decision paid dividends.
"There was big pressure on us, but the manager has trusted him and he has repaid that," he added.
"He's very confident, that's great to see. A lot of the young players who come through are a little bit timid and shy, but he looks comfortable and he's confident in his own ability, which you need to be.
"It was a brave move to play him in the circumstances and we're delighted it's paid off."