Roberto Martinez's team were leading 1-0 when the 23-year-old Costa Rica international collapsed in agony after a first half challenge on Stevenage's Simon Heslop.
He needed lengthy treatment from medical staff before eventually being stretchered off.
Everton team-mates Leighton Baines and Kevin Mirallas were visibly distressed after going over to check on Oviedo.
"A real negative today is the unfortunate incident and we need to give Bryan as much support as we can now," Martinez said, who confirmed the double fracture after the fourth round tie.
"He's had a terrific season and I've been very impressed with him. He's an important player in the group and he deserves a speedy recovery.
"The injury to Bryan affected everyone. Those type of injuries leave a sick taste."
Everton later tweeted: "Bryan has broken two bones in his left leg. He will remain in hospital over the weekend and have an operation on Monday."