Club record signing Pepe put Arsenal ahead in the 10th minute of their third game under their former player by lashing into the net after good work down the left flank from Sead Kolasinac.
Pepe later hit the post in a dominant first-half display from Arsenal before defender Sokratis deservedly doubled their lead in the 42nd minute by pouncing on some hesitant defending from the visitors following a corner.
Arsenal comfortably saw out their advantage to record a first league win in five games which took them up to 10th in the standings on 27 points after 21 games.
Ole Gunnar Solskjaer’s side remain fifth on 31 points after missing out on the chance to close the gap on fourth-placed Chelsea, who drew 1-1 at Brighton earlier in the day and are five points above United.
United were searching for a third consecutive win but were out-fought by Arsenal and rarely looked capable of staging a revival, even after Mason Greenwood and Andreas Pereira came on and injected some energy into their attack.
Arsenal XI: Leno, Maitland-Niles, Sokratis, Luiz, Kolasinac, Xhaka, Torreira, Pepe, Ozil, Aubameyang, Lacazette
Manchester United XI: De Gea, Wan-Bissaka, Lindelof, Maguire, Shaw, Matic, Fred, James, Lingard, Rashford, Martial