Manager says Goodison Park is a difficult place to get a victory and feels his team will continue to improve on last season's seventh place finish following an impressive win
Marouane Fellaini was the hero for the Toffees, scoring the only goal of the game and starring in a superb display, and speaking after the match, Moyes was understandably proud of his team and keen to enjoy the victory.
"It was a great result, great for the crowd. Goodison was at its best tonight and that was partly because the players performed and their attitude and commitment," he told Sky Sports.
"It is a great start, but only a start. You don’t get carried away, but it isn’t very often you beat Manchester United. So I'll enjoy it, the players will enjoy it but we will move on quickly."
The Scot also warned that his team will be a force to be reckoned with this season and insisted Goodison Park would be a difficult ground for teams to visit.
"I thought the team that finished last season did well and I said if I could keep that team we would continue to do well," he added.
"We've added one or two to it, but who knows? The Premier League is a tough league. As I alluded to earlier, I don't think anyone was considering Everton tonight, there was too much attention on United. I think a lot of people forgot Everton aren't a bad team and Goodison is a really difficult place to come and win."
The game also saw Robin van Persie make his Manchester United debut, but he could not find his way onto the scoresheet, and Moyes said the pre-match hype had not distracted his players.
"We weren't making any headlines before this match tonight, it was all about Manchester United, and rightly so, but you mustn't forget Everton aren't a bad side either," he stated.
"Our defence did really well tonight because when we got the goal as expected United put us under lots of pressure.
"If you lose concentration or turn off for a minute they are likely to punish you but I think we defended really well tonight and we caused them some problems with the ball."