Cheltenham made a bright start and Darren Carter had the first opportunity of the game but managed to scuff his effort wide when in a good position.
However, that would prove to be the Robins best moment of the first half as Nickia Jelavic gave Everton the lead in the 12th minute, the Croatian sliding in to poke home the rebound after Marouane Fellaini’s shot had hit the post.
Nine minutes later it was 2-0 after referee Kevin Friend pointed to the penalty spot for Alan Bennett’s foul on Fellaini.
From the resulting spot kick Leighton Baines stepped up to convert, sending goalkeeper Scott Brown the wrong way.
Jelavic and Baines both had chances to extend Everton’s lead before the break as they cruised to the interval 2-0 up.
Things got worse for the League Two side four minutes after the break when Leon Osman danced past one defender to score following Baines’ cut back.
The home fans were given something to cheer about in the 51st minute when Russell Penn pulled one back following Kaid Mohamed’s defence-splitting pass.
Everton restored their three goal advantage seven minutes later though when Seamus Coleman rounded off a fine counter-attack with a deft lob over Brown.
Fellaini then completed the rout in the last minute of normal time when the Belgian midfielder side footed home Victor Anchiebe’s cross.
David Moyes’ side will face either Bolton or Sunderland in round four.