The confirmation of his appointment came after Pellegrini said he had a verbal agreement to move to northwest England, after three years in charge at Spanish club Malaga.
A statement on the club's website mcfc.co.uk said the 59-year-old Chilean had joined on a three-year contract and would take up his role from June 24.
Pellegrini told City TV: "Everyone knows if you arrive at a club like Manchester City you need to win titles.
"I think Manchester City have invested a lot of money to improve the squad every year and try, every year, to win titles. "They were not successful in winning some titles but I am sure we are going to win - maybe the Premier League, the Champions League. We will try to win a lot.
"My first message is to tell all the supporters they will enjoy the season. I am sure they will enjoy the way our team will play. "We will play an attractive, offensive play."
Manchester City won their first league title since 1968 in 2011/12 but although they finished runners-up last season to rivals Manchester United, the campaign was viewed as a disappointment.
Italian Mancini was sacked at the end of the season after City's shock FA Cup final defeat to Wigan Athletic and following failure to progress beyond the group stage of the Champions League for the second successive year.
City chairman Khaldoon Al Mubarak, who has ploughed millions into the club, said of Pellegrini's appointment: "Manuel is a hugely experienced and successful manager with a proven track record.
"We have been greatly impressed throughout the selection process by his philosophy, his attitude and his commitment to the long term development of Manchester City. "I am delighted that he has joined us."
Chief executive Ferran Soriano described Pellegrini as "a very experienced coach with a recognised ability to get the most out of his players and build cohesive teams.
"He shares the club's approach to football and our ambition to achieve on-field success, coordinating with the wider football support teams to ensure natural progression from the Academy to senior level."
Pellegrini has more than 25 years' of managerial experience in South America and Spain and has taken Malaga and Villarreal to the latter stages of the Champions League.
He spent a single season in charge of giants Real Madrid and is renowned for his man-management ability, an area the club had concerns about under Mancini.
Pellegrini said: "I think all my experience will help me at Manchester City.
"With Malaga I reached the quarter-final of the Champions League, with Villarreal also and the semi-final. All my experience will be very useful for me. "I am sure we are going to have a very good season.
"I can tell all the fans of Manchester City I work very hard. I am very persistent in what I want to reach. "I am sure, 100 per cent, I will not have any problems (coming to) the Premier League. I will adapt without any problems."
Pellegrini's appointment at City came shortly after Malaga named former Real boss Bernd Schuster, as his successor.