Oviedo was carried off the pitch after 22 minutes of the match at the Lamex Stadium following a challenge with Simon Heslop.
With the game still in progress, Everton announced on their official Twitter account that the Costa Rica international had been "taken to hospital with a suspected broken leg".
The 23-year-old, who has made 13 appearances for the Merseysiders this season, could now face a race against time to be fit for his country's FIFA World Cup campaign.
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 next year.
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