Oviedo's leg look like it had been snapped in two following a challenge by Stevenage midfielder Simon Heslop in the 18th minute of the 4-0 win at the Lamex Stadium.
Several players turned away in horror at the severity of the injury when it happened.
Everton manager Martinez revealed Oviedo had been taken to hospital and the Spaniard fears a long spell on the sidelines for the Costa Rican.
"Bryan was trying to win the ball but unfortunately I think he has picked up a double fracture, the tibia and the fibula," the Everton manager said.
"He is in hospital now. The medical department will make a decision now on whether surgery is needed.
We will find out over the next 24 hours."
Oviedo stretched his left leg out in an attempt to take the ball from Heslop from behind and the Stevenage midfielder ended up kicking the Everton man's left leg rather than the ball.
The force with which Heslop swung his leg meant a break was inevitable.
Martinez insisted Heslop should not be blamed for the injury.
"It was a very unfortunate action, nothing malicious," the Spaniard said.
Martinez did not put a time frame on the Costa Rican's recovery, but it would be a huge surprise if he played again this season.
The Everton boss appears to expect the 23-year-old to miss World Cup, where Costa Rica are due to play England.
Martinez, who is already without seven first-team players through injury, lamented losing the in-form Oviedo to such a terrible injury.
"It gives you a really bad taste in the mouth," the former Wigan boss said.
"You won't meet a nicer human being than Bryan and this was becoming his season.
"He has shown the quality that he has recently and it's one of those moments where you feel terribly sad for him because he was in good form and the World Cup was just around the corner for him but now we have to make sure he gets fully recovered and he comes back as important player as he has been for the club recently."
The timing of Oviedo's injury could not be worse given that Martinez's team are due to face Liverpool in the Merseyside derby on Tuesday.
Everton's injury list already includes Seamus Coleman, Antolin Alcaraz, Ross Barkley, Steven Pienaar, Gerard Deulofeu, Darron Gibson and Arouna Kone before Saturday's win.
Phil Jagielka came off at half-time, but Martinez said that was a "precautionary" move and the Spaniard expects his centre-half to be fit for Tuesday.
The injury to Oviedo aside, Everton's trip to League One Stevenage proved to be an enjoyable one.
Steven Naismith scored after five minutes to settle any early nerves and the former Rangers man added a second later in the first half.
Second-half substitutes Johnny Heitinga and Magaye Gueye completed the rout over Stevenage, who beat Newcastle at the Lamex Stadium three years ago.
"Stevenage were very brave from the start and we had to show great concentration," said Martinez, who won the FA Cup with Wigan last season.
"We had to refocus after the injury. It was a real test. They did whatever they needed to do to win the game and that is a really important skill if you want to be successful in this competition. I was really pleased and I am looking forward to the draw."
John Stones cleared an early Lucas Akins shot off the line, but other than that Stevenage created little.
Boro manager Graham Westley said: "We had a massive early chance but mentally we didn't show enough strength.
"It was a heavy defeat, but the goals we conceded were soft.
"If you make defensive errors like we did you won't stay in any game."
Oviedo later took to Twitter to express his gratitude to all Everton fans and other well-wishers.
He wrote: "Thank you everyone at this difficult moment. You all are in my heart."