Vrsaljko joins for a reported initial fee of €7 million (£6m/$8m), with that figure rising to €25m (£22m/$29m) when including the purchase price.
Vrsaljko started six of Croatia's seven World Cup matches this summer, including the final, as the Baltic nation finished runners-up in Russia.
Inter have found a replacement at right-back for Joao Cancelo, who joined Juventus in a €40 million (£35m/$46m) transfer in June.
Meanwhile, Atleti moved quickly to find a replacement of their own for Vrsaljko.
Arias, 26, joins on a five-year deal for a reported €13m (£11.6m/$15.2m) fee after starting all four of Colombia's matches at the World Cup in Russia.