The Man United captain had cancelled out Nacho Monreal’s opener but a late Danny Welbeck winner sealed a 2-1 victory for Arsenal.
United have now gone 11 years without winning the FA Cup and it is the only domestic competition Rooney hasn’t won.
Louis van Gaal’s side would have faced either Bradford City or Reading if they had won and Rooney said United would have had a great opportunity to win the competition if they had beaten Arsenal.
Speaking about the result, Rooney said: ”I want to win the cup for and with the team; it’s not about me wanting to win the FA Cup.
“That’s why it’s so sad for us to get knocked out; we knew we’d have a great opportunity if we’d won on Monday night.”
The result against the Gunners was doubly frustrating for United as they made a number of crucial mistakes in the match.
Antonio Valencia gifted Arsenal their two goals with a pair of calamitous errors while Angel di Maria was dismissed for receiving two yellow cards in the space of a minute. His first, for diving, was swiftly doubled after he tugged at the shirt of referee Michael Oliver.