The visitors, who were without the injured Paul Pogba, actually began brightly, but the game took a different turn on eight minutes when Nicolas Pepe steered in his fifth goal of the season after Sead Kolasinac's cross was deflected to him.
That led to a first half in which the hosts were in control and Pepe hit the post before they doubled their lead when his corner was smashed in from close range by Sokratis Papastathopoulos.
Ole Gunnar Solskajer's side came into the game with one loss in their previous nine games, and although they improved after the break, they rarely tested Arsenal goalkeeper Bernd Leno.
It was characteristic of a stop-start season in which they are yet to win three Premier League games in a row.
The defeat leaves them fifth in the table, five points behind Chelsea, who drew with Brighton earlier in the day.
Victory for Arsenal ended a run of seven home games without a win in all competitions and lifted them to 10th place, four points behind United.
But they remain closer to the relegation zone than the top four.
Pepe peps up Arsenal
There has been evidence of a lift in Arsenal's three performances since Arteta was appointed, but without a win, questions remained as to whether he was the right man to take over from former boss Unai Emery, given his lack of experience as a manager.
Arteta was just an onlooker in the stands for the lifeless draw at Everton, but that was followed by a 1-1 draw with Bournemouth on Boxing Day and a crushing late defeat by Chelsea at home last Sunday.
This was a different game altogether, though, as Arsenal put the pieces together. With Granit Xhaka, who has been linked with a move to Hertha Berlin, restored to the midfield, the Swiss and Lucas Torreira were quicker to the tackle than their opponents.
Mesut Ozil covered 11.54km, more distance than any other Arsenal player.
While record £72m signing Pepe was at the heart of Arsenal's attacking endeavour as they were roared on by their fans.
After scoring early on, he sent United left-back Luke Shaw halfway down the Holloway Road with a sharp turn before setting up Pierre-Emerick Aubameyang, who sliced a volley over.
The Ivorian then fed Alexandre Lacazette, who somehow hit his shot out for a throw-in from about six yards out, before Torreira went close from another Pepe pass.
The hosts had to withstand greater pressure from Solskjaer's side in the second period but in contrast to the Chelsea defeat, when they conceded twice in the last seven minutes, Arteta's side managed the game well.
Their only concerns were injuries to Kolasinac and Lacazette, who were both taken off in the second half.
Manchester United improved after the break, but it would have been hard not to do so after a poor first half.
Marcus Rashford had tested Bernd Leno in the first minute, but once they went behind to Pepe's opener, they struggled to match Arsenal's superior energy in midfield.
Jesse Lingard, who returned to the starting line-up, was hardly involved while Nemanja Matic could not get to grips with Ozil, Torreira or Xhaka, who were often one step ahead.
It was another game where Solskjaer's team missed the influence of Pogba, who has not started a game in three months. After the game, the United boss said the French midfielder had picked up an ankle injury which would need an operation, keeping him out for "three or four weeks".
Prior to the game, it had been hinted that Pogba would be fit to play.
United did lift their game once Lingard was substituted, with his replacement Andreas Pereira almost making an instant impact as he hit the side netting.
There were also chances for Fred and substitute Mason Greenwood, but without the injured Scott McTominay, and the ongoing problems with Pogba, United looked short on creativity in midfield and need to find solutions quickly should they want to maintain a top-four challenge.
Man of the Match - Nicolas Pepe
Nicolas Pepe scored, hit the post and set up several good chances, which made his 62nd-minute substitution surprising. Arsenal finally beat top half team.
This was Arsenal's first Premier League win this season against a team currently in the top half of the Premier League (P10 W1 D4 L5).
Nicolas Pepe has scored all his five goals for Arsenal in all competitions in London - four at the Emirates and one at London Stadium.
Arsenal have scored eight goals via corners in the Premier League this season, two more than any other team.
Manchester United have now lost four of their past five away Premier League visits to Arsenal (W1).
Arsenal host Leeds on Monday, 6 January in the FA Cup third round (kick-off 19:56 GMT), while Manchester United travel to Wolves on Saturday, 4 January (kick-off 17:31) in a repeat of their quarter-final defeat last season.