On the back of a successful week with England, the United captain secured victory with a coolly-taken 85th-minute chip over Arsenal substitute goalkeeper Emiliano Martinez.
Rooney’s strike added to a comedy own-goal by Kieran Gibbs that saw the left-back deflect Antonio Valencia’s cross into his own net after a mix-up with goalkeeper Wojciech Szczesny, who was injured in the process.
Olivier Giroud, on as a second-half substitute, replied deep into stoppage time for Arsenal, but United held on to record a first competitive away win since April 5.
The victory took Louis van Gaal’s side back into the Champions League places, 13 points behind leaders Chelsea.
Arsenal manager Arsene Wenger was left to rue an afternoon of wasted opportunities and inspired goalkeeping by United’s David de Gea, as well as injuries to Szczesny and Jack Wilshere.
Arsenal produced a lightning start at the Emirates and should have been out of sight in the first quarter of the match.
Ex-United striker Danny Welbeck was presented with a sight of goal after Alex Oxlade-Chamberlain had been allowed to run in-field, but he was denied by Paddy McNair.
Welbeck then headed over in the ninth minute from a pinpoint Oxlade-Chamberlain cross.
Seconds later, United left-back Luke Shaw appeared to turn his ankle as he cleared the ball under pressure by Oxlade-Chamberlain.
The England international hobbled on, before he was replaced in the 16th minute by Ashley Young.
By then, United goalkeeper De Gea had made saves from Wilshere, twice, and Welbeck.
The second of De Gea’s saves to deny Wilshere came after he had made a poor clearance, but the Spain international made amends with a smart save.
– Wilshere forced off –
Van Gaal’s side finally offered an attacking threat in the 21st minute, but Rooney ballooned a free-kick horribly over the crossbar.
There was a flashpoint just before the half-hour mark when Wilshere was lucky to remain on the pitch after apparently aiming a headbutt at Marouane Fellaini.
After pushing Fellaini in the back, Wilshere was frustrated by referee Mike Dean’s decision to award a free-kick to United.
As the midfield pair squared up, Wilshere seemed to plant his head into Fellaini’s chest, but his relative lack of height in comparison to the towering Belgian may have spared him punishment.
Having weathered Arsenal’s explosive start, the dust-up appeared to spur United into redoubling their attacking efforts.
Angel di Maria curled a shot wide of the left-hand post, Calum Chambers produced a superb block to stop Rooney’s goal-bound effort, and Chris Smalling miscued a volley following a corner.
Wilshere eventually did depart the fray, on 55 minutes, after failing to shake off an ankle injury following a fierce challenge by McNair.
United took the lead within 60 seconds, and in comical fashion.
Szczesny clattered Gibbs as he came for Young’s deep cross and when Valencia smashed the rebound back into the area, Arsenal’s left-back diverted the ball into his own net.
Prior to the opener, United had not registered a shot on target.
Szczesny was forced off after taking a blow to his side and Martinez took over the gloves.
Arsenal pushed for a leveller, but they were undone on the counter-attack as Di Maria supplied Rooney, who raced clear and chipped the ball over Martinez.
Giroud’s blistering late effort from the edge of the box — on his return to the side after a broken leg — set up a grandstand finish for the 60,074 spectators in attendance, but United hung on.