The England international doubled his side's advantage after a Kieran Gibbs goal had broken the deadlock - and the visitors held on despite Olivier Giroud's late strike
The hosts dominated for most of the opening hour at Emirates Stadium but found United goalkeeper David de Gea in excellent form.
The same could not be said for De Gea's Arsenal counterpart, Wojciech Szczesny, whose failure to deal with a ball into the box led directly to Kieran Gibbs' 56th-minute own goal on the end of Antonio Valencia's drilled cross, before the Pole was then forced off injured.
United then made light of high-profile defensive absentees by closing out their first competitive away win of the Louis van Gaal era, a triumph sealed by Wayne Rooney on the counter-attack five minutes from time.
It could have been three late on when Angel di Maria missed with a one-on-one breakaway opportunity, before Olivier Giroud gave Arsenal something to cheer with a superb finish on his return to the side after a fractured tibia.
There were, though, injuries to Jack Wilshere and Szczesny as Arsenal's miserable record against United was extended to only one win in 15 attempts - and the gap to Premier League leaders Chelsea was extended even further.
Danny Welbeck shook off a hamstring problem to lead the Arsenal attack against his former club, while Arsene Wenger preferred Mikel Arteta to Mathieu Flamini in midfield.
Van Gaal's latest response to United's defensive injury crisis was to name Tyler Blackett and Paddy McNair in a re-shaped back three.
McNair produced an important early block to deny Welbeck and the England international headed over in the ninth minute as Arsenal began in the ascendancy.
Wilshere and Welbeck were then able to test United goalkeeper De Gea as his new-look defence allowed too much room on the edge of the penalty area.
In the 15th minute, De Gea atoned for a terrible clearance by keeping out Wilshere's resulting one-on-one opportunity when the midfielder should have left him with no chance.
Left wing-back Luke Shaw's departure with an ankle problem further increased United's defensive uncertainty and the alert De Gea had to race from his goalmouth to deny Alex Oxlade-Chamberlain.
Wilshere was fortunate to remain on the field after thrusting his head towards Marouane Fellaini in a 29th-minute clash – a reflection on his frustration at Arsenal losing their early control of the match.
Di Maria was central to United’s improvement, flashing a speculative strike narrowly wide and then setting up Rooney, only for Arsenal full-back Calum Chambers to make a vital 38th-minute block.
An excellent pass with the outside of his foot from Oxlade-Chamberlain sent Welbeck racing into the United box from the left-hand side early in the second half, but De Gea was equal to the task once more.
A misunderstanding between McNair and Fellaini had Wilshere racing through on the United goal again soon afterwards, but a challenge involving both United players resulted in ankle damage and an early end to the Arsenal man's afternoon.
To compound Wenger's worries, his team fell behind in the 56th minute.
Poland international Szczesny failed to deal with a cross from Shaw's replacement, Ashley Young. In doing so he knocked Gibbs to the floor and the defender put the ball through his own net when Valencia drove a cross back into the area.
Szczesny injured himself in the process to bring about an unplanned Premier League debut for substitute goalkeeper Damian Martinez.
In-form forward Alexis Sanchez placed a header into De Gea's hands as Arsenal sought a response and Santi Cazorla, on for Wilshere, had a rasping effort saved.
Arsenal committed increasing numbers forward late on, meaning Rooney was able to race onto Di Maria's pass before lifting the ball beyond Martinez, and his team up to fourth in the table.
Di Maria then passed up a glorious chance to add a third on the break and Giroud, back in the side after his fractured leg, smashed a powerful shot into the top corner to make United sweat midway through an elongated nine minutes of stoppage time, but there was to be no comeback.