The 30-year-old has joined the Nerazzurri on a short-term contract until the end of the 2023-24 season, with the option of an additional 12 months.
Klaassen becomes Inter's 11th new permanent signing during the summer window, following Alexis Sanchez, Marcus Thuram, Benjamin Pavard, Davide Frattesi, Yann Sommer, Juan Cuadrado, Kristjan Asllani, Yann Aurel Bisseck, Francesco Acerbi and Raffaelle Di Gennaro to San Siro.
The Dutchman departs Ajax following a successful three-year spell in his homeland, having returned to the club where he spent 13 years between 2004 and 2017 during the earliest days of his career.
While the Amsterdam giants lost their way domestically last season, Klaassen still managed 10 goals and three assists from 46 matches in all tournaments, occupying a central midfield and number 10 role throughout the season.
Klaassen has also represented Werder Bremen and Everton during his distinguished career, and the Nerazzurri new boy has 10 goals to show from 41 matches in a Netherlands jersey.
The midfielder could potentially make his debut for Inter when they host Fiorentina in Sunday evening's Serie A showdown.