The Milan club laid out pounds 10.4m for a double-Dutch deal that included Bergkamp's club-mate, Wim Jonk, a defensive midfielder who was valued at pounds 3.3m. The pair will join Inter next season, on three-year contracts.
Bergkamp, 23, and coveted by every side with the money to match their ambitions, had been most strongly linked with Juventus and Barcelona, but he said: 'Inter met all my demands. The most important thing for me was the stadium, the people at the club and their style of play.'
Inter won their 13th Italian title in 1989, though in recent seasons they have not been top dogs even in their own city. Milan, building phenomenal success around their Dutch trio of van Basten, Ruud Gullit and Frank Rijkaard, lead second-placed Inter by nine points in Serie A and appear destined for their second successive title. Last week they Inter 3-0 in an Italian Cup quarter-final.
Bergkamp and Jonk came to prominence last season as Ajax won the Uefa Cup, beating two Italian clubs, Genoa and Torino, along the way. Bergkamp has scored 12 goals in just 19 internationals while Jonk has played six times for his country.
The governing body of international football, Fifa, has agreed to the new Japanese professional league, which starts in May, adopting sudden-death finishes to drawn games. Two 15-minute periods of extra time will be played, followed by a penalty shoot-out if necessary.