Inzaghi had confirmed last week that he was leaving his role as Lazio coach, and will now take over from Antonio Conte.
Conte agreed to the mutual termination of his contract just weeks after he led Inter to their first Serie A title since 2010.
Inzaghi has been at Lazio since joining as a player in 1999, winning Serie A, the Coppa Italia three times and a UEFA Super Cup as a striker before hanging up his boots in 2010.
He started coaching in the capital club's youth sectup before taking the first-team job in April, 2016 after Marcelo Bielsa left his role at the club after just two days.
Inzaghi guided Lazio to the Coppa Italia and two Italian Super Cup successes, as well as leading them to the round of 16 in their first Champions League campaign in 13 years this past season.
He will now take over from Conte, who masterminded Inte's title success -- the club's 19th overall -- breaking Juventus' nine-year stranglehold on the trophy.
Inter made the high-profile signings such as Romelu Lukaku, Christian Eriksen and Achraf Hakimi during Conte's tenure and their arrivals spurred the club to great success. But a bleak financial outlook and a prospect of transferring players made Conte not keen to work with a "downsized" project.
Inter were also part of a group of 12 clubs that had planned to launch a breakaway European Super League, before backing out. The club are said to be upwards of €500 million in debt, due to both the coronavirus and ambitious spending.
The majority owner, the Chinese retailer Suning, has been dealing with the effects of a state-sponsored bailout and has been actively seeking new club investors.
Reuters contributed to this report.