Marouane Fellaini came off the bench to score the winner in the 115th minute and spare the blushes of Roberto Martinez's Premier League side.
Luke Freeman's left-footed strike in the first half had put the League One club ahead and raise the prospect of an upset on Merseyside.
But Steven Naismith set up Gerard Deulofeu to score on his Toffees debut and pull one back shortly after the interval, before Fellaini's goal then helped set up a third-round trip to Fulham.
Everton had made eight changes and handed out debuts to goalkeeper Joel Robles, defender John Stones and midfielder Deulofeu while Arouna Kone was given his first start.
Ross Barkley was a brilliant driving force throughout for Everton and was always willing to shoot. He twice threatened in the opening minutes with one effort deflected and another flying over before David Gray somehow denied Kone from a yard out.
Deulofeu, another constant menace to the Stevenage defence, went close when he shot across goal from a tight angle and the Spaniard then set up Leon Osman, only for the midfielder to miskick.
Everton were made to pay for their profligacy 10 minutes before the break when substitute Oumare Tounkara raced onto a long ball behind the home defence. He pulled back for Smith, whose fluffed shot rolled perfectly for the inrushing Freeman to fire into the roof of the net.
The Toffees got the goal they deserved on the stroke of half-time as Deulofeu underlined his ability with a shimmy that took out two defenders and then passed the ball into the far corner.
After spurning a host of chances, Everton fans showed frustration as Deulofeu shot straight at Day after another quick break six minutes from time. Barkley then shot over from the edge of the area and Gray's well-timed tackle denied Deulofeu another chance.
Stevenage almost snatched victory in injury time as Greg Tansey - grandson of former Toffees player Jimmy Tansey - forced Robles to tip over from 25 yards.
Fellaini entered the action to loud applause six minutes into injury time and made his presence felt in the second period of extra-time but was brilliantly denied from 12 yards by a low Day save as Stevenage struggled to clear.
However, the Belgian settled the contest as the spectre of penalties loomed, pouncing after Roarie Deacon failed to clear a cross.