Bayern Munich midfielder Costa, who played in the club's penalty shoot-out win against Borussia Dortmund in the DFB-Pokal final last week, was ruled out of the Copa America due to a left-thigh injury.
Brazil coach Dunga elected to replace him with Orlando City midfielder Kaka, who has contributed five assists and scored twice in seven appearances in the MLS this season.
Kaka made his Brazil debut under Luiz Felipe Scolari in 2002 at the age of 19, and has gone on to play 91 times for the national team.
Brazil are looking to win a ninth Copa America title at the special Centenario edition of the tournament, held in the United States next month.
Dunga and Co. begin with a clash against Ecuador on June 4, before facing fellow Group B opponents Haiti (June 8) and Peru (June 12).