In the first half only a superb display by Manuel Almunia kept United at bay, but the Gunners improved in an even second period and Nicklas Bendtner almost headed a late equaliser.
United make a whirlwind start to the match and, in only the second minute, Darren Fletcher's cross found Wayne Rooney whose looping header was clawed away by Almunia.
When Kieran Gibbs slipped Cristiano Ronaldo then drilled over a low cross that evaded everyone before Almunia denied United again in the 17th minute.
The Spanish goalkeeper made a superb save, getting down well to keep Carlos Tevez's close-range shot out after good play by O'Shea.
However, the deadlock was broken from the resulting corner which wasn't properly dealt with. Michael Carrick's cross was deflected into the path of the unmarked O'Shea who was allowed to ram his shot into the roof of the net.
Arsenal's first shot on target didn't arrive until the 28th minute when Cesc Fabregas forced Edwin van der Sar to keep his 25-yard shot out after tricky play by Theo Walcott.
On the half-hour mark, only another brilliant save by Almunia stopped United doubling their lead. Ronaldo's header from Tevez's pinpoint cross seemed certain to burst the net, but the goalkeeper somehow kept the ball out.
In the second half Arsenal kept possession for long spells as they got a toe-hold in the game, but they didn't really have a cutting edge.
After a quiet opening to the second period, United substitute Ryan Giggs made an instant impression in the 69th minute when he set up Ronaldo who rifled in a fierce shot that crashed back off the crossbar.
Carrick's 30-yard drive was inches wide with Almunia struggling to get across his goal and Giggs had the ball in the net only for the veteran Welshman to be denied by the linesman's flag.
However, Arsenal almost grabbed an equaliser with four minutes left when substitute Bendtner got his head to Fabregas' free-kick only to send the ball over the bar.