United climbed to fifth in the Premier League table after their victory, as Robin van Persie's first-half header saw them extend their unbeaten run to nine matches in all competitions.
League leaders Arsenal struggled in the first half and, despite an improved showing after the break, their winless run at Old Trafford - which stretches back to 2006 - continued.
"This was a big result for us today," Moyes told Sky Sports. "Arsenal have been in terrific form this season and we had to play very well ourselves.
"I thought we started the game well. We were very good in the first 20 to 30 minutes. I thought in the end, our whole determination, effort and good play helped us in the second half when we were under more pressure.
"I thought we did that (apply pressure) much better in the first half. I thought we gave them a bit too much room in the second half.
"In the second half I think we had the best opportunities but we didn't take them. But overall, three important points."
Moyes also revealed that defender Nemanja Vidic had been sent to hospital after being hit in the head by the hip of team-mate and goalkeeper David de Gea in an innocuous collision late in the first half.
"He's gone to hospital. I've only just seen it myself," Moyes added.
"It's a terrible clash. It's an accident. Hopefully he will be OK. I'll find out more from the medics later."
Wayne Rooney swung in the corner for Van Persie's goal against his former club and the England forward produced an all-action display, with his work-rate and tenacity standing out.
Only one player – Arsenal's Aaron Ramsey – covered more ground than Rooney during the match, and the striker felt his side's defensive effort was crucial.
"It's a great result. Arsenal are a good team," Rooney said.
"We knew we had to win today at all costs. I think it's the toughest we've been to play against, to break down. We defended really well. Thankfully Robin got the goal and we got the three points.
"We knew we couldn't afford to lose today. We knew Arsenal couldn't get 11 points ahead of us. We knew a victory would put us right back in there. We're in a great position."
