Arsenal remain sixth, six points behind Manchester City, who occupy the fourth and final Champions League berth, but they are now just two points below fifth-place United and retain games in hand on both Manchester teams.
Wenger overcame Mourinho’s Chelsea in the 2015 Community Shield, but this was the first time he had got the better of his old foe with league points at stake.
While the odds remain stacked against Arsenal, they will approach their final four matches with hope intact that they may yet squeeze into the Champions League for the 20th season running.
For United, the Europa League now looks even more like their best hope of securing Champions League football for next season.
Mourinho made no fewer than eight changes to the team that won 1-0 at Celta Vigo last Thursday, an advantage United will hope to defend in this Thursday’s semifinal return leg at Old Trafford.
Matteo Darmian, Ander Herrera and Henrikh Mkhitaryan were the only players to keep their places from United’s win in Vigo.
The shake-up yielded a full debut for 19-year-old Axel Tuanzebe, who lined up at right-back but was charged with the task of following Alexis Sanchez wherever he went.
With Darmian shadowing Mesut Ozil on the other half of the pitch, Mkhitaryan and Juan Mata – back after a seven-week lay-off with a groin injury – found themselves playing as auxiliary wing-backs.
The visitors had two early sights of goal, Wayne Rooney heading over and Anthony Martial drawing a near-post parry from Petr Cech, but it was Arsenal who made most of the running.
Persisting with their new 3-4-2-1 shape, the hosts were allowed to advance deep into United territory and chances soon arrived.
XHAKA’S STROKE OF FORTUNE
Aaron Ramsey drew a smart save from David de Gea in the ninth minute, his drilled effort from an inside-right position obliging the Spaniard to drop to his right and palm the ball behind.
Welbeck twice threatened, only to be thwarted by blocks from Phil Jones and Chris Smalling, while Alex Oxlade-Chamberlain warmed De Gea’s palms with a deflected effort from 25 yards.
A lapse in concentration from Rob Holding almost gifted United a goal just beyond the half-hour.
His blind pass from wide on the Arsenal right wing rolled straight into the path of Rooney, but Cech raced from his line to produce a sprawling block.
The second half was trundling along at the same, fairly pedestrian pace before two goals in the space of three minutes dropped victory into Arsenal’s lap.
They took the lead in the 54th minute thanks to a huge stroke of fortune as Xhaka’s speculative 25-yard strike smacked into Herrera’s turned back and looped over the despairing dive of De Gea.
Before United had had time to clear their heads, it was 2-0.
Oxlade-Chamberlain, who is thriving at right wing-back, shaped a cross into the box and Welbeck met it on the edge of the six-yard box with a header that cannoned into the net via the underside of the bar.
Mourinho sent for reinforcements, Jesse Lingard and Marcus Rashford joining the fray in place of Mkhitaryan and the hapless Herrera.
But Arsenal continued to boss possession and United came no closer to reducing the arrears than a Rooney free-kick – one of several long-range efforts by the visiting skipper – that Cech turned behind.
Mourinho gave a debut to 20-year-old midfielder Scott McTominay in the closing stages, but there was little else for United to cherish.