Gasperini, sacked by mid-table Genoa in November after a shaky start to the season, was not Inter's first choice after a number of names including former Argentina coach Marcelo Bielsa turned them down.
They were interested in Andre Villas-Boas but baulked at the price of his Porto release clause and he instead joined Chelsea.
"Gian Piero Gasperini will be the new Inter coach," a club statement said.
"President Massimo Moratti, currently out of Italy, confirmed 'his full satisfaction with the choice' and, together with the entire club, wishes Gasperini 'the success that brought Inter the title of world champions'."
Gasperini will sign a two-year deal.
Judging by fan reaction, the appointment has not met universal approval with Inter supporters hoping for a more high-profile name.
Leonardo, who replaced the sacked Rafa Benitez in December after the Spaniard had succeeded 2010 treble winner Jose Mourinho, led Inter to the Italian Cup in May but they meekly surrendered their Champions League title in the quarter-finals and lost their Serie A crown for the first time in six years.
AC Milan, who were previously managed by Leonardo, took the scudetto off the city rivals who were runners-up.
BADLY BACKFIRED
Leonardo had long been a director at Milan and took some persuading to take over from Carlo Ancelotti as coach in 2009.
His one season as Milan boss ended in a third place Serie A finish but he left after falling out with owner Silvio Berlusconi only to make a shock return at Inter.
The managerial lifestyle did not ultimately suit the Brazilian though and Inter's gamble badly backfired after being given warnings from his time at Milan that he preferred being a sporting director in a less hand-on and less stressful role.
Gasperini inherits an ageing side with rumblings that top players Samuel Eto'o and Wesley Sneijder could leave while preisdent Moratti has shown a recent tendency to tighten his purse strings ahead of the advent of financial fairplay.
UEFA's fairplay rules state clubs are banned from owing money to other clubs, players, tax and social security departments - a move aimed to curb reckless spending by clubs and encourage them to live within their means.
The 53-year-old Gasperini was formerly the Juventus youth team coach before managing lower league Crotone and then taking charge of Genoa in 2006 while they were in the second tier.
They won immediate promotion to Serie A under his stewardship and finished fifth and qualified for the Europa League in 2009 thanks in part to Diego Milito and Thiago Motta, two players he will now rejoin at Inter.
The start to last season was lacklustre though and ambitious Genoa axed him after spending large sums on several new recruits.
