Spaniard Benitez, who last week ended his six-year spell at Liverpool, has penned a two-year deal with the European, Serie A and Italian Cup champions.
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"Inter welcome Rafael Benitez," the club said in a statement, adding the coach would hold a news conference on Tuesday.
Mourinho quit for Real Madrid in the wake of Inter's Champions League triumph last month, having become disillusioned with Italian football.
The 50-year-old Spaniard knows plenty about Serie A having visited for European games with Liverpool and faced Inter's city rivals AC Milan in both his Champions League finals.
He already speaks decent Italian, which went down well with the country's demanding reporters, and his love of tactics will endear him to the many Serie A watchers who view football as more of a chess game.
Doubters have pointed to his shaky record in the transfer market in the last two seasons, with some of Liverpool's buys not being seen as sufficient class for the Anfield faithful.
Repeating an unprecedented Italian treble will be incredibly difficult, if not impossible given Inter's ageing squad will be a little older and will have lost their guiding light in Mourinho.
Another factor counting against Benitez will be the re-emergence of AS Roma who would have beaten Inter to last season's Serie A title had they not had a mediocre start.
Juventus, long linked with Benitez before appointing former Sampdoria boss Luigi Del Neri as coach, are rebuilding after a dreadful season similar to Liverpool's and look likely to be more of a force next term.
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