Oviedo, 23, sustained the distressing injury during his side's 4-0 win at third-tier Stevenage in the FA Cup on Saturday and now seems likely to miss Costa Rica's FIFA World Cup™ campaign.
He remained in hospital over the weekend before undergoing surgery on Monday.
The head of Everton's medical team, Danny Donachie, told the club website: "Bryan's surgery was a big success and he is in a comfortable and stable condition."
He added: "It was a clean break and that means that there are no pieces of bone. Sometimes when there are small fragments of bone in the area, it can be difficult to deal with.
"But these are clean breaks, so they will put the bones back into position and it will heal well from there."
"Ever since it happened, I've been thinking about the World Cup," the left-sided player told the club website.
"I have to remain calm, but, in the hands of God, I will be ready for it."
He added: "You learn from experiences like this. You take knocks in life, but the key is to be strong and to fight back."
Oviedo's leg was badly broken as he attempted to block a shot by Stevenage's Simon Heslop during the first half of the game, leaving some of his team-mates in tears.
"I have tried to stay strong and positive, but I don't want to watch the video of the incident," Oviedo said.
"I don't think I will ever want to see it."