The Costa Rica international now looks certain to miss the FIFA World Cup as a result of the injury he sustained in the first half of the Merseyside club's 4-0 fourth-round victory at the Lamex Stadium.
Oviedo went down in agony after a challenge with Simon Heslop that resulted in a free-kick for the League One side, and he was subsequently taken to hospital for scans.
Everton have since revealed that the 23-year-old will need to undergo surgery on Monday.
The Premier League club tweeted: "Bryan has broken two bones in his left leg. He will remain in hospital over the weekend and have an operation on Monday."
Everton were leading 1-0 when the incident occurred and went on to progress to the fifth round courtesy of a Steven Naismith double and second-half goals from John Heitinga and Magaye Gueye.
But Oviedo's injury left Everton manager Roberto Martinez deflated.
"A real negative today is the unfortunate incident (Oviedo's injury) and we need to give Bryan as much support as we can now. " he told BT Sport.
"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.
"We always looked a threat on the counter and a threat in front of goal. It was as good as it gets really in terms of a performance. It was a good cup tie, but the injury to Bryan affected everyone. Those type of injuries leave a sick taste."
Oviedo has made 13 appearances for the Merseysiders this season and scored the winner at Manchester United last month after also finding the back of the net in the previous Premier League game against Stoke City.
The left-sided player joined the club for an undisclosed fee from Copenhagen in 2012.
He has been an important figure for his country since making his debut in 2010, playing nine times as they qualified for the World Cup finals in Brazil.
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