Steven Naismith scored twice for the visitors in the first half to give them a commanding lead over their lower-league opponents.
Here, Sports Mole takes a closer look at the action between the two teams at the Lamex Stadium.
On target: 6
On target: 7
Was the result fair?
There can be no doubt that Everton fully deserved the win thanks to a dominant display pretty much from start to finish. They took their chances when they got them and looked every bit the Premier League side that they are against lower-league opposition.
The hosts started gallantly with plenty of gusto, but that wore off as the match went on. They struggled to live with the quality of Everton and some sloppy defending cost them in the end. That said, you would not have guessed that this was a team at the bottom of League One. They showed a lot of heart, and manager Graham Westley will have been content with what he saw. More of that and they could have a fighting change of staying up this season.
The Toffees never really had to get out of second gear to book their place in the fifth round. Their passing was crisp, while the movement of the front players gave Stevenage lots to think about. Defensively there was enough there to please manager Roberto Martinez ahead of a tough Merseyside derby in midweek. This was far from the quicksilver Everton that we've been used to seeing in recent weeks, but they still made light work of their opponents.
Sports Mole's man of the match
Steven Naismith: The Everton forward was a wily presence all night for the visitors as his cunning runs and clever movement caused Stevenage plenty of problems. The Scot was rewarded for his display with two well-taken goals.
Stevenage could have taken the lead early on when Lucas Akins had time in the box to pick his spot, but the forward could only shoot straight at John Stones on the line.
Stevenage: The Boro continue their fight for League One survival on Tuesday as Crewe Alexandra visit the Lamex Stadium.
Everton: The Toffees return to action on Tuesday when they travel to local rivals Liverpool in the Premier League.