The only goal of the game arrived just before the half-hour when Robin van Persie rose to head home Wayne Rooney's corner.
Moyes admits he had not enjoyed too many 'good weekends' since taking charge of United but this was one of the best so far.
"We have not had many good ones, but this was a big result for us today - Arsenal have been in terrific form this season and we had to play well to get the result," Moyes told Sky Sports.
"I thought we started the game well, we were very good in the opening 20-30 minutes in the game.
"I thought our whole determination, the effort, commitment, good play, helped us in the second-half when we were under more pressure.
"It's another step in the right direction for us.
"We have got a lot of big steps to take here and it is going to take a while for me to get it all the way I'd like it to be.
"I am not singing and dancing because we won the game, but I am thrilled because the boys put in a great performance."
Moyes admitted that they had been looking to restrict Arsenal's game as much as possible.
"I thought we did that much better in the first-half, I thought we maybe give them a little too much room in the second-half but we restricted them in the first-half and we played from that, but in the second-half we had the best opportunities but did not take them but overall three important points."
The result was a huge one for United on a weekend which saw the likes of Chelsea, Manchester City and Tottenham also drop points - but Moyes believes the Premier League has a lot more twists and turns to come.
"I didn't think in the Premier League that anyone had run away with it," he said.
"I think there are a lot of ups and downs and I think it is a close run league this year and there will be a few shocks and we have had one or two ourselves in the early part of the season.
"I am pleased we are showing a bit more consistency and momentum and hopefully we can continue that."
Moyes had special praise for the inspirational Wayne Rooney and his strike partner Van Persie.
He said: "Great players score great goals in big games and he (Van Persie) is a great player.
"I have got to say, his sidekick today played great as well, Wayne - the two of them were fantastic."