Chivu was transferred for 16 million euros with Inter defender Marco Andreolli moving the other way in a shared ownership deal between the two clubs, Roma stated.
"I am going to Inter: it is a life choice," Chivu said. "I would like to be remembered as a good lad. I have spent four marvellous years here, full of satisfaction."
Inter had pulled out of negotiations earlier this month before an 18 million-euro move to Real Madrid collapsed because of the 26-year-old defender's salary demands.
Barcelona had also expressed an interest in versatile Romania captain Chivu, one of Serie A's top defenders last season when Roma finished second behind Inter and won the Italian Cup.
Roberto Mancini's Inter finally went back in for the former Ajax player this week and have wrapped up a deal despite Chivu starting pre-season training with Roma.
Roma had already bought Brazil centre back Juan from Bayer Leverkusen to replace Chivu, who is expected to partner Italy's Marco Materazzi in the centre of the Nerazzurri's defence.
Inter, who have agreed to play a friendly with Roma within the next three seasons as part of the deal, flew to London on Friday for a pre-season tournament at Arsenal.
Chivu was not included in the party.