The 23-year-old - who signed for the club from Copenhagen in 2012 - made his first Premier League start of the season in Saturday's 4-0 mauling of Stoke City, and marked the occasion with a superb goal just before the hour mark.
This came over a year after Oviedo's only previous league start for the Merseysiders, and the Costa Rica international - who normally plays in midfield - expressed his determination to take advantage of regular left-back Baines being sidelined for up to six weeks with a broken toe.
"I am very happy to have scored my first goal in the Premier League and also to play my first game at left-back," he told Everton's official website.
"I'm also happy that the team played well today. This is a very good chance for me. Leighton is a great player but I have waited a year and a half for this chance and I am so happy to play again.
"I think the coming games are a good opportunity to play for the team and make the most of my chance.
"The team is a strong team and a good group. They are all good guys and good players."
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