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 Barclays 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."