Oviedo's left leg looked to have been snapped in two after he was caught by Stevenage midfielder Simon Heslop as he was attempting a shot on goal.
Everton's players turned away in horror as the midfielder lay stricken on the floor at the Lamex Stadium. The Costa Rican was taken to hospital and now looks set for a lengthy spell on the sidelines.
Everton, who were 1-0 up at the time thanks to Steven Naismith's goal, did not let the incident affect them as they strolled to a comfortable 4-0 win.
Naismith found the net again before substitutes Johnny Heitinga and Magaye Gueye finished the rout against the Sky Bet League One side.
Roberto Martinez will be pleased to have avoided an upset at Stevenage, who beat Newcastle at the Lamex Stadium three years ago, but there will be deep concern for Oviedo, who has impressed recently for the Merseysiders.
A serious injury to one of his key players is the last thing the Everton manager needs ahead of Tuesday's Merseyside derby, especially as he already has seven first-team players out through injury.
Writing in his programme notes, Stevenage boss Graham Westley said his team would go on the attack from the start, and Stevenage lived up to that promise early on.
With four minutes gone Everton failed to clear Luke Freeman's cross and the ball dropped to Lucas Akins, who would have put the home side ahead had John Stones not stopped his shot on the line.
Everton responded in the best way possible. The Toffees raced straight up the other end through Kevin Mirallas, who drew a top save from Chris Day. McGeady picked up the loose ball and squared for Naismith, who turned home from six yards.
Stevenage came back into the game momentarily. Former Chelsea man Filipe Morais swung a deep cross in the direction of Jimmy Smith, but Stones came to the rescue again.
Then, in the 18th minute, Oviedo suffered his horrific injury. The Costa Rican stretched out his left leg attempting to take the ball from under Heslop's feet.
Oviedo caught the full force of Heslop's right boot as he attempted to shoot and the midfielder fell awkwardly, twisting his leg.
Replays showed the midfielder's leg was bent almost at a right angle as he lay in agony.
Several players rushed to Oviedo's aid, but some turned away in horror at the severity of the injury before waving towards the bench for assistance.
After just over three minutes of treatment, Oviedo was carried off the pitch on a stretcher to a standing ovation from all corners of the ground.
Joel Robles had to be at his best to save from the resulting free-kick, which was taken by Freeman.
The former Wigan stopper pulled off another top save, diving to his right to deny Morais from a free-kick moments later.
Just as Stevenage started to come back into the game, Everton doubled their lead. Leon Osman picked the ball up on the halfway line and passed towards Naismith. A moment of hesitation from Boro captain Jon Ashton proved costly as the ball slipped through to Naismith and he slotted the ball under the goalkeeper.
After the break Martinez replaced Phil Jagielka with Johnny Heitinga and within 10 minutes the Dutchman made it 3-0.
Mirallas played an audacious overhead kick across the box and Heitinga swept the ball home via a deflection from Stevenage's Darius Charles.
Any hopes of an upset had gone, but Stevenage did not give up.
In-form Francois Zoko nudged a half volley just wide of Robles' goal with 20 minutes to go.
Soon after it was 4-0, though. Osman charged at the Stevenage defence and laid the ball through to Gueye, who dodged a couple of challenges before beating Day.