The Spanish champions had been linked with a move for the France international throughout close-season, with coach Diego Simeone publicly praising the player over the weekend.
And Atletico announced on Monday that they have reached an agreement with Sociedad for Griezmann, who will put pen to paper on a contract at the Vicente Calderon after undergoing the "relevant medical examinations".
Sociedad also revealed the news on their official website, and expressed their gratitude to the 23-year-old - who came through their youth system - for his contribution to the club.
"Real Sociedad would like to thank Antoine Griezmann [for] the professionalism and dedication shown during his years spent at our club, while wishing you the best for your personal and professional future," their statement read.
Griezmann joined Sociedad 13 years ago, making his first-team debut in 2009, and scored 53 goals in 202 appearances for the San Sebastian club.
His form last season, which saw him score 16 times in 35 La Liga matches, earned him a place in Didier Deschamps' France squad for the World Cup.