The 33-year-old agreed a four-year contract to join City as new manager Pep Guardiola continues his major overhaul of the 2014 Premier League champions.
"Claudio is a fantastic goalkeeper and an excellent addition to our squad," Guardiola told the club website.
"He has experience and great leadership qualities and is in the prime of his life. He is a goalkeeper I have admired for a number of years and I'm really happy he is now a City player."
Bravo moved to Barcelona from Real Sociedad in 2014 and enjoyed great success, winning the Spanish title in 2015 and 2016 and the Champions League in 2015.
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The Spanish champions said Bravo had moved for an initial 18 million euros plus a further two million in add-ons. They signed his replacement Jasper Cillessen from Ajax Amsterdam earlier on Thursday for 13 million euros.
"It is not easy to leave a club like Barcelona where I had two fantastic years, but the opportunity to work with Pep Guardiola was too good to refuse," Bravo said.
"Now I will challenge the other great goalkeepers the club has and together I hope we can win many trophies."
The keeper, who has 106 caps for Chile and won the Copa America in 2015 and 2016, became City's 11th signing of the transfer window.
Goalkeeper Geronimo Rulli, defenders Pablo Mari and John Stones, midfielders Ilkay Gundogan, Nolito, Aleksandar Zinchenko, Aaron Mooy and Marlos Moreno along with forwards Leroy Sane and Gabriel Jesus have all joined the club.
Bravo could make his City debut at home against West Ham United on Sunday.
His arrival looks likely to push England's Joe Hart closer to the exit door at the Etihad Stadium, although no official bid has been received for the 29-year-old.
(Reporting by Ed Dove, editing by Ed Osmond)
