After Manchester United's 2-1 win over Liverpool earlier in the day, the pressure was on Mancini's men to deliver or end the weekend 10 points down in the Barclays Premier League title race.
The visitors were handed an advantage when referee Mike Dean sent off Arsenal defender Laurent Koscielny for wrestling Edin Dzeko to the floor after just 10 minutes.
Although the Bosnian saw his penalty saved, via the post, City - who had last won in the league at Arsenal in 1975 - took command with first-half goals from James Milner and Dzeko.
Arsenal, whose defeat saw them lose more ground on the top four, regrouped after the break, but were not able to find a way back into the game - even when City captain Vincent Kompany was sent off with 15 minutes left for a two-footed tackle on Jack Wilshere.
Mancini however was unhappy with the decision and confirmed after the match that the Citizens will look to get the red card rescinded.
"We will appeal, because it is impossible that we can lose one player for three games for nothing. It is not red card, it is nothing," said the Italian.
"He went into the tackle, anticipating the opponent and took the ball before.
"I don't know how it is possible to get a red card like this. It was no foul, no red card.
"He went in with one foot, that is a tackle. If you want to be right and just, we can say what has happened - if it was a red card, I would say, but it was not a red card, because he anticipated Wilshere. Totally, 100%."
Kompany took to Twitter to give his side of the story, accompanied by a photo which appeared to show one of his feet being on the floor as he made the tackle.
"No grudges against the referee, I understand the difficulty of the job," he said.
City remain seven points behind United, and Mancini insists all is not yet lost.
"At this moment, United play very well, but the season is long. It is important for us to be there, close with them, and we will arrive the moment they lose points," he said.
"We have the derby on April 6 and we need to arrive there with five or four points behind, it is okay."