United bounced back from a difficult fortnight by returning to winning ways with a 2-0 success over Cardiff.
Mata enjoyed a commendable debut, playing a part in the build-up to both goals, which were scored by the fit-again Robin van Persie and Ashley Young.
The win moves United to within striking distance of sixth-placed Everton, but the home side were far from convincing in the way they dispatched the team that currently sit bottom of the Barclays Premier League.
Cardiff had United under pressure at the start of the second period and, despite Mata's energy, it was only towards the end of the match that the reigning champions started to turn on the class.
"My mood has improved because I know I have some really good players coming back, but I still want to be more convincing," the United manager said.
"I want to win better and play much better. Don't think because we have won tonight that that means I have a smile and I am happy.
"I am happy for the three points but I have a long way to go in terms of the vision I have of how I want my team to play."
Mata's arrival at United's Carrington training ground aboard a helicopter was like something out of a Hollywood blockbuster.
His performance at Old Trafford was slightly more low-key, but nevertheless the Spaniard's addition to Moyes' starting XI gave United a bit more flair and tempo.
The World Cup winner covered all four corners of the pitch in an energetic display and the way in which he lofted a 50-yard diagonal ball to Patrice Evra in the build-up to United's first was very impressive.
It was Van Persie's return from a seven-week thigh injury that proved to be the most telling, though. The Dutchman evoked memories of his title-winning form last term when he pounced on a rebound from his original header to beat David Marshall in the sixth minute.
With substitute Wayne Rooney also returning to action after a four week layoff, Moyes thinks he has the ingredients to turn United into world-beaters again.
"The results with them (Rooney, Van Persie and Mata) will get much better," the former Everton manager said.
"We need to get them in the right physical condition now to make that happen.
"To see Robin van Persie and Wayne Rooney back on the touchline would have been enough to give everyone an uplift but then to have Juan Mata as well... Everyone in the crowd gave him a fantastic reception and I think the crowd saw he was creative in the early part of the game. He made our best chances in the opening 20-30 minutes so I was really pleased."
Moyes has been promised the cash to fund a significant rebuilding job at United, but it appears the majority of the work will be done in the summer.
Any United fan hoping for another marquee signing before the transfer window looks set to be disappointed.
When asked whether he thought he would bring in anyone else before Friday's deadline, the Scot said: "I would say no. I have said no, I don't think I will, but if something happens I will soon let you know."
Moyes was just as coy when it came to talk that Rooney is on the verge of extending his contract at Old Trafford.
"I am sure we will keep you up to date with that as well," he said.
United fans gave Ole Gunnar Solskjaer a rousing reception when he took his place in the away dugout.
The Norwegian, 11 years a player at United, he was unfortunate to come to Old Trafford just as Moyes had Van Persie, Rooney and Mata all available again.
"It was a good time to come to Old Trafford," said a smiling Solskjaer, whose memorable goal gave United their dramatic Champions League win 15 years ago.
"But I thought we did well against them.
"I am pleased with our overall performance but we lacked that goalscorer who wants to score a goal, we lacked that 'oomph'."
Solskjaer hopes Kenwyne Jones, who completed his move to Cardiff from Stoke on Tuesday will be the man to provide that "oomph".
"He will help us. It is just a shame we didn't get the papers signed in time for him to play today," the Cardiff manager said.
Solskjaer said a deal to sign left-back Fabio from United had been completed, but a loan deal for Wilfried Zaha has still not gone through.