“I think it’s another step in the right direction for us,” he said.

“But we’ve got a lot of big steps to take here. It’s going to take a while for me to get it all the way I’d like it to be.”

He added: “We know we’ll get a few bloody noses along the way. I think the Premier League has shown that it’s not just Manchester United who are getting a few of them.”

Going into the international break, just six points separate Arsenal and eighth-placed City, and Moyes says the campaign is living up to expectations following a close season of widespread upheaval.

“At the start of the season, we all said [this would happen] because of the changes in managers’ jobs,” he said.

“More likely, the reason Arsenal have started the season so well is because they’ve had continuity. They’ve had a manager who’s been there for a lengthy period.”

Former Arsenal striker Robin van Persie scored the game’s only goal, heading in a 27th-minute corner from strike partner Wayne Rooney, and Moyes praised both players.

“Great players score great goals in big games,” said the Scot.

“He’s a great player. And I’ve got to say, his sidekick played great as well today, Wayne. I thought the two of them were both fantastic, the way they played.”

The only troubling aspect of the game for United involved captain Nemanja Vidic, who had to go off at half-time after having his head jerked back in an awkward mid-air collision with goalkeeper David de Gea.

He was taken to hospital after the game, but Moyes played down the extent of his injury.

“I’ve had a chance to see it [the incident] and it’s quite a horrific-looking one,” he said.

“As far as I know, he’s OK, he’s conscious. He’ll get a scan, but as far as we know, we hope he’s going to be OK.”

Arsenal goalkeeper Wojciech Szczesny played on despite apparently being left dazed following a similar collision with United’s Phil Jones earlier in the game.

Tottenham Hotspur were criticised for not withdrawing goalkeeper Hugo Lloris after he was momentarily knocked out at Everton last weekend, but Arsenal manager Arsene Wenger said his medical staff had not raised any concerns about Szczesny.

“I listen to the doctor,” he said. “If he says to me he has to come off, he comes off. No matter if we are only seven men on the pitch, I’ll do it.”

Wenger attributed his side’s defeat – their first in the league since the season’s opening day -- to a “nervous” first-half display and the “mistake” of allowing United to score from a set-piece.

Arsenal do not play again until November 23 and Wenger, whose side lead Liverpool by two points, said the international fixture window had fallen at an inconvenient time.

“The international break is not welcome because when you lose a game like that, you want to play straight away again,” he said.

“We have some players who go to play massive games under massive pressure and [who] will not come back after the break refreshed. They will come back maybe exhausted.”

Wenger claimed that referee Michael Oliver had got “a few” decisions wrong during the game, but he was reluctant to elabourate.

“I don’t want to talk too much about that,” he said. “I need to buy some Christmas presents and all these statements cost money.”