Robin van Persie scored against his old club as United overcame the Barclays Premier League leaders 1-0 in a tense clash at Old Trafford.
The result lifted United to fifth in the table and within five points of the Londoners in spite of the difficulties new manager Moyes encountered in the opening months of the season.
Asked if that was his best performance, Moyes said: "Yes, especially in the first half.
"A resilient job was required in the last 20 minutes but overall it was a good three points against a very much in-form Arsenal team.
"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 played down suggestions the win for the champions had sent out a warning to the rest of the competition.
He said: "I don't know if it puts out any statement.
"Everybody for years has known how good Manchester United have been.
"My job and the team's job is to make sure that we do that again.
"We know that we are going to get a few bloody noses along the way.
"But Manchester United were champions last year and there's a lot of people written them off very quickly - which is fine - but hopefully come the end of the season we will not be too far away from it."
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."
Rooney himself told Sky Sports 1: "We knew we'd have to win today at all costs. I think we defended really well and thankfully Robin got the three points and we got the three points, and I think it was a great effort from everyone."
"We knew we couldn't afford to lose today and we couldn't let Arsenal get 11 points ahead of us, and I think the other results today and yesterday...we knew a victory would put us right back in there and we're in a good position.
"We worked on free-kicks and corners as Arsenal have a lot of smaller players in the team and that could give us the opportunity to get a goal, and thankfully it's paid off."
Arsenal boss Arsene Wenger felt his side showed signs of nervousness early on and were unable to impose themselves on the game.
He said: "It is disappointing to lose a game the way we lost it.
"At the start of the game we didn't play our game. We were nervous, insecure and we gave them a little bit of comfort by making a mistake at the first set-piece, or the second.
"After that we lost too many balls and rushed our game in the first half.
"In the second half when we played our game we played in one half and they counter-attacked a few times.
"We didn't find the opportunity to equalise.
"We had the opportunities but we missed always, by a few inches, to push the ball into the net.
"They were well focused to defend and catch us on the break and that happened once or twice in the game.
"It is disappointing as we had the opportunity at least to get a draw out of this game."
The result prolonged Arsenal's poor record against United at Old Trafford, having not won at the ground since September 2006.
Wenger felt that probably contributed to his side's poor start.
When asked why his team were nervous, Wenger said: "Maybe with the fact we haven't won for a long time here and it was a big game and, once again, it is a tight game that has been decided on a set-piece.
"That is what we will keep from this game.
"We knew this could happen but after that we had a go and gave everything until the last minute. This team has a great heart.
"I feel sorry for my players tonight."
Wenger felt referee Michael Oliver made some wrong decisions but was unwilling to elaborate.
He said: "I need to buy some Christmas presents and these statements cost too much money now."
He also did not want to blame the virus which had affected a number of his players in the past week and ruled out Per Mertesacker and Tomas Rosicky.
He said: "That can happen the whole season, it doesn't mean you have to lose a game.
"They declared themselves fit so they play. When they play, they play. There is no excuse. I felt physically we were all right."