Bolasie, 27, injured his right knee during the Toffees' 1-1 draw with Manchester United on 4 December.
Everton manager Ronald Koeman says the DR Congo international underwent surgery on his torn meniscus on Thursday.
"Now he has to wait for nine to 10 weeks before surgery on his anterior cruciate ligament," the Dutchman said.
Koeman says Bolasie, who joined Everton from Crystal Palace in a £25m deal in August, is likely to be sidelined "for a long time" and will definitely miss his country's Africa Cup of Nations campaign, which starts in Gabon next month.
Everton winger Yannick Bolasie will undergo a second operation on his knee injury in the New Year.