Romelu Lukaku wasted a couple of big chances for United either side of Xhaka's goal as United, unbeaten in 12 top-flight matches before today, slipped below their opponents into fifth place.
Arsenal's top-four destiny is not quite in their own hands, though, as Chelsea - held to a 1-1 draw by Wolverhampton Wanderers earlier today - can draw level on points if they win their game in hand against Brighton & Hove Albion.
The Gunners edged in front through their only on-target attempt of the first half as Xhaka's swerving shot somehow found its way past De Gea, who put too much weight on his left leg and was unable to get across to his right.
Either side of that long-range opener United, 3-1 winners here in the FA Cup on their most recent visit less than two months ago, came close as they hit both the crossbar and the post.
Lukaku was guilty of wasting a glorious chance just a couple of minutes before Xhaka made the breakthrough, sending a left-sided cross from Luke Shaw against the crossbar from less than six yards out.
United were the side on top as the first half edged towards its conclusion as Lukaku, boasting a hat-trick of braces heading into this match, forced Leno to make a save after being slotted through by Rashford.
The visitors were on top for the opening 15 minutes of the second period, only denied a leveller by the quick reactions of Leno as he kept out Lukaku's low shot with his legs.
For all United's probing, however, they were unable to find a way through and the game was put out of their reach with 20 minutes to play.
Alexandre Lacazette raced into the box and went to ground under small contact from Fred, allowing Aubameyang to redeem himself for his North London derby penalty miss by tucking this latest one past De Gea down the middle.
A fan ran onto the field during the celebrations and appeared to shove United defender Chris Smalling, potentially landing Arsenal in hot water with the Football Association.
Some poor defending from Victor Lindelof let Lacazette in moments later, though he pulled his shot wide of the target with just De Gea to beat.
Rashford headed an Ashley Young cross right at Leno from the game's final chance, meaning just a second defeat in 18 matches for United since Solskjaer took over from Mourinho.
Arsenal, meanwhile, are now within a point of bitter rivals Tottenham Hotspur thanks to a third home league win in four meetings with the Red Devils.
ARSENAL (4-3-1-2): Leno; Sokratis, Koscielny, Monreal; Maitland-Niles, Ramsey, Xhaka, Kolasinac; Ozil (Iwobi 77'); Lacazette (Nketiah 86'), Aubameyang (Suarez 80')
MANCHESTER UNITED (4-4-2): De Gea; Young, Smalling, Lindelof, Shaw; Dalot (Martial 71'), Fred, Matic (Greenwood 80'), Pogba; Lukaku, Rashford