Sinclair, 23, excelled at Swansea after struggling to break through at Chelsea. He was an integral part of the side that earned promotion to the top flight in 2011 and also impressed on Swansea's opening season in the Barclays Premier League.
The move is one of the more surprising of the summer transfer window.
Wright, who has made 467 career appearances and won two England caps, joined League One Preston earlier in the summer but left a week later citing homesickness.
The 31-year-old Maicon received an undisclosed fee.
Garcia, 25, underwent a medical Friday and signed a deal to become City's fifth capture of a busy transfer deadline day. City gave no details of the length of contract while reports claimed the player was likely to have cost around 16 million pounds ($25.4 million).
The former Real Madrid player will effectively replace Nigel de Jong in the squad after the Holland international left for AC Milan.
Serbian defender Nastasic moves to England from Fiorentina, with City misfit Stefan Savic heading in the opposite direction, along with a fee of 9 million pounds ($14.3 million). The 19-year-old impressed for the Viola last season, and is seen to able to fit into the new City formation of 3-5-2.
Wright made his name as a teenager with Ipswich, for which he played more than 300 games over three spells.
"We need a goalkeeper with experience and Richard Wright is good for us," Mancini said. "Sinclair is looking forward to playing in the Champions League following his move to Eastlands."
He joined Arsenal in 2001 and has also played for West Ham, Southampton and Sheffield United.
"I'm just glad everything is settled and that I'm now a City player," said Sinclair. "There were times when I thought it might not happen so I'm relieved that I'm here and I can't wait to get started. Playing alongside some of the best players in the world is exciting for me and when you see two Champions League fixtures against Real Madrid on the horizon, it brings it all home."
Maicon is familiar with Mancini having worked under him for two seasons at the San Siro.
He was available for a possible cut-price deal as he was in the final year of his contract with Inter Milan. Mancini denied an interest in the player last week but was a little more forthcoming when he spoke to preview this weekend's game against QPR.
The Italian said: "We are talking about one top player in Maicon, because he has experience."
Maicon has previously played for Cruzeiro and Monaco and won 66 caps for his country. He will join up with his new teammates when he returns from international duty next week.
Information from Press Association was used in this report.