The 23-year-old was linked with a big-money move to a host of clubs but Primera Division champions Atletico have won the race for the France international.
Atletico announced on their official website: "Atletico Madrid and Real Sociedad have reached an agreement for the transfer of Antoine Griezmann.
"The young French footballer will sign a contract with our teams once he passes his medical."
Neither club has confirmed the transfer fee but reports in Spain suggest Atletico have met the 30million euro (£24million) buy-out clause in Griezmann's contract.
Sociedad said in a statement: "The player has the permission of Real Sociedad to undergo his medical in Madrid.
"Real Sociedad want to thank Antoine Griezmann for the professionalism and dedication he has shown during his time at our club, and at the same time wish him the best for the future."
Barclays Premier League trio Liverpool, Arsenal and Tottenham were among those sides believed to have been keen on Griezmann, along with French outfit Monaco.
However, it is Atletico who have secured his services as the champions look to rebuild after losing several key players from their title-winning side, with Diego Costa, David Villa, Filipe Luis, Adrian Lopez and Thibaut Courtois among those to leave.
Griezmann scored 53 goals in 202 games for Sociedad, who he joined as a 13-year-old, while he has made nine appearance for France including five at this summer's World Cup.