Ansaldi, who has been capped by Argentina on three occasions, is capable of playing on either side of the defence.
Atleti announced his arrival on Friday, after the 27-year-old passed a medical.
Jose Luis Perez Caminero, the club's sporting director, said: "Ansaldi is a player who's going to help us increase the group's competitiveness.
"He's a full-back who can play at both sidelines, with a great defence level and offers many variables on the attack.
"He will bring plenty with his versatility. We are very pleased with this agreement."
Ansaldi's arrival helps compensate for the recent departure of Filipe Luis, who left Atleti last month to join Chelsea.