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."
@solskjaer99 is an idiot. Period. I've been reading his ramblings about Moyes not being good enough to win anything at Everton as a reminder why he isn't good enough for United. Tell me one thing, you tool, was there any chance in hell that any manager in this world would have won cups and leagues with that Everton outfit whilst competing with Sir Alex's United and the oil baron clubs? Don't give me Wigan winning the FA Cup as an example, ask any Wigan fan whether they'd have preferred to stay in the top flight or to win the FA Cup, most would've preferred the former. Moyes achievement has been converting Everton from a club which battled relegation to a contender for the Champions league on a shoe string budget. As far as I'm concerned that is exactly what United want, a manager who can help a club over achieve just the way Fergie was over achieving with our squad which was probably the 3rd or 4th best in the league for the past few years all the while giving youth a chance. That constitutes a great manager, someone who manages to get more than the sum of the players he's left with. If you want proven players and managers dependent of big buys to succeed all the time like Mourinho (the guy who signed Samuel Etoo...), then support Chelsea or Man City you plastic tw*t. And yes that is exactly what you are, a plastic. Saying that if Moyes proves you wrong and wins leagues with us will only make you happy is only a defensive response you've readied for when you'll be proven wrong. Moyes has all the qualities to the perfect manager for Manchester United in keeping with our traditions. We need less 'fans' like you around though to help him settle in with us.- utd_till_i_die