Moyes saw some calm brought to a turbulent few days when Javier Hernandez's solitary strike at the start of the second half proved enough to defeat Liverpool 1-0 in Wednesday night's Capital One Cup third-round tie at Old Trafford.
The victory earned United a home last-16 encounter with Premier League strugglers Norwich after a kind draw - one of the few things to have gone well for Moyes since he succeeded Sir Alex Ferguson in the summer.
Moyes' sceptics had a field day following Sunday's 4-1 hammering at Manchester City, which led to claims the Scot is incapable of securing wins in the biggest matches.
For the moment, they have been silenced, as Liverpool striker Luis Suarez's return from a 10-match suspension ended in disappointment.
And, after being told he had now made the best start of any new United manager since Matt Busby in 1946, Moyes claimed he will eventually do it for good.
"Everybody seems to be picking a stat out - I have not won here or I have not done that," he said. "That is one of the good stats.
"I have not had that many good ones. But that is one of the better ones.
"It is is not frustrating. I will sort it. I will fix it. I will turn it round. I have no doubt about that."
Moyes has every reason to look at the world with a good deal more optimism than he did at the weekend.
After making eight changes - and naming Wayne Rooney captain - the United boss saw his team perform with more attacking conviction than they have done in most of their more important matches this season.
"It certainly was," said Moyes, when asked whether his reaction at the final whistle was one of relief. "It was a big game for us. We had lost at Anfield a few weeks ago and at the Etihad on Sunday and we wanted to put that right.
"Every game everybody is either a first for something or it is a monkey off my back. It is going to be the way, isn't it? Liverpool have started the season well, so to get a result was good.
"It certainly wasn't a risk playing the team I did. The players are all internationals and first-team players. There are no first team and reserves in this squad. Some of the performances were excellent tonight and they had to be."
Phil Jones, Nani and Shinji Kagawa all caught the eye, as did goalscorer Hernandez, who was netting for the first time this season.
"Sometimes in the box it is about movement, and losing your defender," Moyes said. "Javier did that. And a lot of credit has to go with the way he finished it as well, the way he adjusted his body. It was a really good goal."
Liverpool manager Brendan Rodgers was satisfied with Suarez's performance but knows there is a lot more to come from the striker.
"I thought Luis was excellent," he said. "Considering he had been out for such a long time he ran himself into the ground, he was always a threat, he put defenders on the back foot with a couple of twists and as he goes on he will get a bit sharper.
"Overall he can be really pleased and I was really happy for him because his contribution was brilliant. He has worked very hard. He obviously tired a bit towards the end but the only way he is going to get that (fitness) is playing in games. But even in the last moments of the game he can still create, distract defenders and open up the space."
Old Trafford was probably not the ideal place for Suarez to make his comeback after five months on the sidelines considering the history he has with the club following his eight-match ban after being found guilty in December 2011 of racially abusing their defender Patrice Evra.
But Rodgers said there was never any doubt the Uruguay international would play against their arch-rivals.
"He is one of the most determined people I've ever met in my life," added the Reds boss. "He is very thick-skinned and he has a wonderful talent. There are no qualms he will go and play anywhere.
"I had spoken to him and he was fine. When you are out for so long not playing when the first opportunity comes along he was never going to be fazed by it."
Rodgers also saw it as an opportunity for the 26-year-old to repay the fans after a testing few months in which he tried to manufacture a summer transfer by claiming Rodgers and the club reneged on a deal to allow him to leave for a team in the Champions League.
"The support they have given him is unheralded. They were brilliant towards him, he has given them everything," he said. "He was clapping to the crowd before the match started so he recognises everything they have given him.
"Now he has got to get his head down and give them that great talent he has and hopefully it will end up in goals for him."
While he was full of praise for Suarez, Rodgers appeared to point the finger at Jose Enrique for losing Hernandez at a corner for the goal.
"It is disappointing. It is an individual thing, it is not a collective," he said. "If you look at the corner you see all the players skin-tight to their markers, no one was free and then Chicarito gets free far too easily. It is not good and at the weekend (in the 1-0 defeat to Southampton) we did a similar thing.
"As an individual you have to be stronger mentally and have a greater desire not to lose your marker. If you do that there is a greater chance you won't concede. It is not something we go into straight after the game but it is something they know."
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