Mancini, one of Inter's most successful managers, returned to the San Siro club for a second spell on Friday after Walter Mazzarri.
Stankovic, now assistant coach at Udinese, won three Serie A titles during Mancini's first stint at the Nerazzurri from 2004 to 2008 and expects the Italian to revitalise the club.
"He knows the environment, and when you have no time to lose, that is something that makes a big difference," the Serbian said.
Inter currently sit ninth in Serie A, 12 points behind leaders Juventus, but in young Croatian midfielder Mateo Kovacic, Mancini has a player he can build a team around, according to Stankovic.
"Inter and Inter fans, they needed this: to find a good feeling with their coach," he added.
"He likes the players like Kovacic: young, with quality and who are not afraid to take responsibility," the former midfielder said.
"He can build [a team] around him. It will be a boon for Mateo and Inter. "
Mancini's second reign will begin with a derby match against AC Milan Sunday November 23.