The 24-year-old France midfielder, glad the protracted battle over his move was over, signed a four-year deal at the world's richest club with local media reporting the transfer fee was around 25 million pounds.
"It's a big relief because it was a long, long negotiation and discussion between the two clubs," Nasri, who was chased for most of the close season, said on the City website.
"Now I'm finally a City player. That's what I wanted since the start.
It was that potential that attracted Nasri to the club with the midfielder saying compatriot and former City player Patrick Vieira had told him it was "the club of the future".
"He's a fantastic player because he has technique and mentality," Mancini said in a video on City's website on Wednesday.
"I knew him when he was playing at Marseille because I followed him when I was at Inter [Milan], we wanted to talk to him five years ago. In the last four years he has improved a lot."
"I was a little bit frustrated because I said to the [Arsenal] manager I wanted to leave," Nasri said.
"So I stayed for the whole pre-season, I wanted to do the pre-season with Manchester so I could know my team mates a little bit more."
Nasri, who scored 15 goals for Arsenal last season, could make his City debut on Sunday when his new club travel to face his old club's bitter North London rivals Tottenham Hotspur.
"He's a player maybe like David Silva. But he can also play in the middle," the Italian, who has hinted he has not quite finished shopping in this transfer window, said.
Under Mancini, who took over as manager in late 2009, they were third last season, one place above Arsenal in City's highest Premier League finish. They also won the FA Cup, their first major trophy in 35 years.
Nasri's exit was another blow to Arsenal who have endured a torrid start to the season, including former captain Cesc Fabregas' departure for Barcelona earlier this month.
