The 29-year-old striker penned a three-year deal with the reigning Serie A champions on Wednesday after the Eastlands outfit agreed a £10m fee with La Vecchia Signora.
The Argentina international leaves the Citizens having won the Premier League title and two FA Cup crowns, while the former Boca Juniors forward scored 73 goals in 138 appearances.
Despite Tévez’s City spell peppered with both on-field and off-field controversies, Barry insisted the former Manchester United man was a key player for the Citizens.
“It will be a massive loss for Manchester City,” the midfielder told Sky Sports News.
“He has had a fantastic few years there and he has been fantastic to play alongside.
“To hear that news is quite sad but he will go and do a great job wherever he goes.”
Former City manager Mark Hughes lured Tévez to the Eastlands outfit from bitter rivals United in 2009, and the Argentine was appointed captain for the 2010/11 campaign by the Welshman’s successor, Roberto Mancini.
However, Tévez fell out with the Italian after refusing to come on as a substitute in a Champions League game at Bayern Munich in 2011, returning to Argentina before featuring again for City six months later.
A statement from City read: “Everyone at Manchester City wishes Carlos the best of luck in Italy and thanks him for his important contributions.”