Robin van Persie's 27th-minute header against his former club settled a cagey match at Old Trafford and lifted United to fifth place, with Arsenal's second defeat of the campaign trimming its lead over second-place Liverpool to two points.
With Man City losing 1-0 at struggling Sunderland and Spurs beaten at home by Newcastle by the same scoreline, just six points separate Arsenal and eighth-place City after 11 matches in what is proving to be a tight, exciting battle for the title.
"What we have learned today is that not only Man United can beat Arsenal but that everybody can beat everybody in this Premier League," Arsenal manager Arsene Wenger said. "It just shows you how much excitement there will be until the end of the season."
City's poor away form continued as Phil Bardsley grabbed the winner in the 21st minute for Sunderland, which started the day level on points with bottom-place Crystal Palace.
City has taken only four points from a possible 18 on its travels.
"I am concerned because we are losing too many points away and, of course, that's very difficult if you don't win away to try to fight for the Premier League," City manager Manuel Pellegrini said.
More doubts were raised about Tottenham's top-four credentials after in-form striker Loic Remy struck a 13th-minute winner at White Hart Lane, with Newcastle goalkeeper Tim Krul preserving the victory with a string of brilliant saves.
"I think Tim Krul was the key to the game," Tottenham manager Andre Villas-Boas said. "He made the difference."
Spurs followed medical advice and chose not to play goalkeeper Hugo Lloris, who is still recovering from being knocked unconscious during a match against Everton last weekend. Lloris also missed the midweek Europa League win over Sheriff Tiraspol, despite having played on against Everton.
In Sunday's other match, Charlie Adam converted a penalty in the sixth minute of second-half stoppage time to salvage Stoke a 3-3 draw at Swansea.
After beating Liverpool 2-0 last weekend, Arsenal was looking to pass another examination of its title credentials but failed to shake off an inferiority complex the Gunners seem to have when they arrive at Old Trafford.
Wenger said his side was "nervous" in the first half, during which Van Persie answered the relentless taunts of visiting fans by heading Wayne Rooney's corner into the top corner. Since leaving north London for Manchester in August last year, Van Persie has scored in all three games against Arsenal.
In a game of few clear-cut chances, Arsenal established control in the second half as United reorganized after losing defender Nemanja Vidic to a head injury sustained just before the break. Vidic was later taken to hospital for a scan.
However, home goalkeeper David de Gea wasn't seriously tested as Arsenal tried to find a way through the United defence.
It was the biggest win for Moyes since taking over from Alex Ferguson this summer and United is now unbeaten in its last nine matches in all competitions after a scratchy start to his tenure.
"I don't think it puts out a statement, I think people have known for years how good Manchester United have been," Moyes said. "My job and the team's job is to try to show we can do that again.
"We know we are going to get a few bloody noses along the way. And I think the Premier League has shown it's not just Manchester United getting them, there are a few others as well."
City can attest to that.
Having scored 12 goals in beating Norwich and CSKA Moscow over the past week, City was expected to ease past Sunderland. But their foibles away from home came back to haunt them, despite dominating possession and territory.
Sunderland remains in the bottom three but seem to be heading in the right direction under new manager Gus Poyet, who also guided the team to a northeast derby victory over Newcastle in his last home game.
"The proudest time is when you are able to keep a clean sheet against a team like Man City," Poyet said. "For a team who did not keep a clean sheet before, it is perfection."Suggest a correction