City were the better side in the first half and took a deserved lead through David Silva after a lightning counter-attack in the 18th minute.
But Arsene Wenger's men responded well after the break and levelled through Flamini, who scored an own goal in the 2-2 draw with Swansea in midweek, after he was set up by Lukas Podolski early in the second period.
He told Sky Sports: "We wanted to win the game.
"It was important for us and for the fans especially after the 6-0 defeat to Chelsea and the draw against Swansea on Tuesday, but unfortunately we came away with a draw.
"We came back at half-time.... we saw in the first half there was the opportunity to create some goals so we came out in the second half and tried our best.
"We had the chance to score a goal and we tried to score another for the victory, but we missed the opportunity."
A bizarre own goal from Flamini in the last minute handed Swansea a 2-2 draw at the Emirates on Tuesday night, and he was pleased to put things right against City.
He added: "It was a pleasure, especially after the own goal against Swansea.
"I'm pleased to score this goal, especially against Manchester City and I'm pleased to score it at home."