Costa has been playing with pain throughout the 2018-2019 campaign, caused by a pin in the fifth metatarsal on his left foot, and travelled to Brazil for surgery this week.
The Spain striker has scored a single goal in LaLiga this season, against Barcelona last month, and has been linked with a move away from Atletico despite only returning to the club in January.
Atletico has not confirmed how long it will be without Costa for, but reports in Spain have suggested he could miss two months of action.
"The intervention consisted in the removal of the osteosynthesis material of the V metatarsal that our player underwent 12 years ago and in repairing the short peroneal tendon," Atletico's statement said.
Costa's injury may provide an opportunity for Nikola Kalinic to shine, the striker having started only two LaLiga matches since signing from AC Milan in August.