The holders scored in the 17th minute when Irish defender O'Shea rifled in a half-volley from eight yards after Michael Carrick [ Images ] had turned Anderson's deep corner back across goal.
United dominated the match and can feel they should be travelling to next week's second leg at the home of their Premier League rivals with a greater advantage.
"From our first-half display we maybe should have had a couple more goals, but we have a lead and kept a clean sheet and we know we are capable of scoring a goal at the Emirates," O'Shea said in a televised interview.
"But it's definitely not over."
Arsenal's only real opportunity of the opening period was a shot from Cesc Fabregas [ Images ] which was sliced well wide of goal.
Cristiano Ronaldo [ Images ] headed straight at Arsenal goalkeeper Manuel Almunia [ Images ] from a perfect cross by Carlos Tevez [ Images ], who was handed a surprise start ahead of Dimitar Berbatov [ Images ].
United forward Wayne Rooney [ Images ] also went close after cutting in from the left flank and sending a curling shot just wide of Almunia's goal.
The pattern continued after the restart and Anderson shot well over following fine work from Ronaldo and Rooney.
A rare opening for the visitors saw striker Emmanuel Adebayor shoot too high after he was picked out by a clever pass from Alex Song.
United went close to doubling their lead when Ryan Giggs [ Images ] sent Ronaldo into space and the Portuguese's thumping shot from more than 25 yards bounced away off the crossbar.
The hosts were nearly made to pay for their failure to add a second goal when Arsenal striker Nicklas Bendtner [ Images ] headed wide from Fabregas's free kick but Arsenal rarely threatened to get a potentially vital away goal.
There was concern for United in the closing stages when England [ Images ] defender Rio Ferdinand [ Images ] left the field clutching his ribs but Alex Ferguson's [ Images ] team held firm to stay on course to reach the final in Rome.