Despite the presence of the TV cameras, willing Stevenage to cause an upset, Everton did a professional job to progress through to the next round, but the game was marred by a serious injury to Bryan Oviedo.
The Costa Rican suffered a double leg break in the 16th minute in a innocuous looking challenge, much to the horror his team-mates.
By that stage Everton were 1-0 thanks to Steven Naismith and the Scot doubled the visitors' advantage just before half-time.
Jonny Heitinga made it three nil early in the second half to kill the contest, allowing Everton to slow the game down and conserve energy ahead of Tuesday's trip to Anfield.
Another substitute, Magaye Gueye, completed the job by sliding home a fourth with six minutes remaining.