The victory at the Emirates Stadium keeps sixth-placed Arsenal's hopes of a top-four finish alive, but they are six points adrift of Manchester City, who currently occupy the final Champions League qualifying slot, albeit with a game in hand.
The win also takes Arsenal within two points of fifth-placed United, who stay four points behind City having played the same number of games as their local rivals.
"Hopefully this result will give us confidence," said man-of-the-match Aaron Ramsey. "I said a couple of weeks ago we had to win every match and unfortunately we didn't do that against Tottenham (in a 2-0 loss last weekend). But we will see."
For all of United manager Jose Mourinho's pre-match bluster about fielding a weakened team, his only concession to youth in eight changes from his midweek Europa League team was 19-year-old defender Axel Tuanzebe, who made his Premier League debut.
Everything else was familiar, with United under orders to produce the sort of bodies behind the ball performance that makes them so hard to beat.
However frustrating the tactics, there is no doubting their effectiveness and the first half finished with Arsenal having spurned a series of chances and the Emirates again jittery.
Twice United defenders had denied Welbeck on the six-yard area while darting runs from the ever impressive Ramsey almost saw him get on the end of pinpoint balls from Aaron Sanchez.
United also had their chances before the break, with Arsenal goalkeeper Petr Cech saving from Anthony Martial.
But after a battling first-half display, the visitors were undone after the interval with Xhaka's 25-metre shot looping up off Ander Herrera to catch out United keeper David De Gea after 54 minutes and then Welbeck adding a second against his former side with a header from Alex Oxlade-Chamberlain's cross.
There was no way back for United and, for once in a troubled season, the Arsenal fans could relax and forget the will he-won't he concerns about whether manager Arsene Wenger will renew his contract. The win could not have been better timed and was Wenger's first over Jose Mourinho in a league game.
For United, the only realistic route to Champions League qualification now lies in winning the Europa League in which they host Spaniards Celta Vigo at Old Trafford in the semi-finals on Thursday holding a 1-0 win from the first leg.
"Yes is all on the Europa league," said Mourinho, "but you cannot say we did not try to win."
(Reporting by Neil Robinson; Editing by Ken Ferris)
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