Despite coming from behind to beat Sunderland on Saturday, the Red Devils are only ninth in the Premier League table after seven games.
But Rooney has urged his team-mates to use the comeback against the Black Cats as motivation in the future, and says the squad are still adapting to Moyes' style.
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"We know as a team it's us who have let him down. We have to pull together for the manager and show the sort of fight we have done here and we'll be fine.
"It will be because it's a new manager with different styles and we have to adapt to them," he argued of the drop in results. "We need to do better for him than we have been."
Adnan Januzaj was the hero for United at the Stadium of Light. On his first start, the 18-year-old scored twice to turn the game around and the England international was thrilled to see the youngster excel.
"He's a great prospect," Rooney enthused. "He takes the ball, he runs at players and he's got great ability. It's great the manager has thrown him into the side, especially with what has happened in the last few games.
"There was big pressure on us but the manager has trusted him and he has repaid that. 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.
"There is a lot of pressure but he seems to thrive on it. We know the ability he has got but you never know how they are going to react when they are thrown into games.
"We need to help the younger players and, if we do that, they'll be moments where they come in and win us the game, as has happened here. The experienced lads need to help them emerge into a team."