Mancini's position had been in doubt because of his reported misgivings over Inter's transfer business.
"I'm very happy to be here, and I hope to see you soon," De Boer, 46, said in a message on Inter Milan's Twitter.
Inter, who were bought by Chinese retail giant Suning in June, were beaten 6-1 by Tottenham in a pre-season match on Friday.
They announced Mancini's departure "by mutual agreement" on Monday. De Boer has signed a three-year contract.
The former Manchester City boss returned to Inter for a second spell as manager in 2014 but was unable to guide them back into the Champions League, finishing fourth last season.
High-level discussions took place with the club's new Chinese owners in New York last month and, after a third pre-season defeat, they have parted with Mancini just two weeks before the new Serie A season.
Former Netherlands international De Boer left Ajax earlier this summer and had been linked with the Southampton and Everton jobs.
He led the Amsterdam club to four successive Dutch titles but lost out to PSV Eindhoven on the final day of last season.
Inter Milan have appointed former Ajax boss Frank de Boer as manager, after parting company with Roberto Mancini.