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.
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