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, who were 1-0 up at the time thanks to Scotland striker 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 three years ago, but there will be deep concern for Oviedo, who has impressed recently for the Merseysiders.
Writing in his programme notes, Stevenage boss Graham Westley said his team would go on the attack and they 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. Aiden McGeady picked up the loose ball and squared for Naismith, who scored from six yards.
Everton doubled their lead when Leon Osman picked the ball up on the halfway line and passed towards Naismith. A moment of hesitation from Stevenage captain Jon Ashton proved costly as the ball slipped through to Naismith and the former Rangers striker slotted home.
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. Six minutes from time it was 4-0. Leon Osman laid the ball through to Gueye, who dodged a couple of challenges before beating Day.