Chivu, who has long been groomed for the role, will be in charge in time for their Club World Cup campaign.
Former Inter defender Chivu, who left Parma less than four months in charge, played as a defender for Inter for seven years from 2007 and was part of Jose Mourinho's treble-winning side in 2010.
He later spent six years coaching their youth teams, from under-14s in 2018 to under-19s in 2024.
"In February 2025, Cristian was given his first opportunity to coach a senior side as Parma... He led the Emilians to safety, with the team picking up 16 points from 13 matches under his guidance," Inter said in a statement.
"Now, Cristian Chivu is beginning the third chapter of his Nerazzurri story. And he's doing so with the values that have always guided him: dedication, sacrifice and a tireless work ethic."
Inter begin their Club World Cup campaign against Mexico's Monterrey on June 17 in Los Angeles.
Inzaghi, who left the club last week after four years as coach, has joined Saudi Arabian side Al-Hilal.
