Nikica Jelavic put Everton ahead in the 12th minute, before Leighton Baines doubled the lead from the penalty spot a few minutes later.
Leon Osman extended the lead just after half-time, but following Russell Penn's strike for the home side, goals from Seamus Coleman and Marouane Fellaini put the Premier League side out of sight.
David Moyes' side will now visit the winners of next week's Sunderland-Bolton replay at the Stadium of Light in the fourth round.
Moyes paid Cheltenham the compliment of parading eight of the line-up which started the 2-1 midweek win at Newcastle in the Premier League.
Cheltenham started brightly and former Birmingham player Darren Carter had half a chance but dragged his shot wide after being found in space by Kaid Mohamed.
However, Everton broke the deadlock after 12 minutes through Jelavic.
Fellaini's curling shot from the right of the Cheltenham box contained too much power for Scott Brown as it crashed against the far post. Jelavic was alert to the rebound and made no mistake for his seventh goal of the campaign.
After 19 minutes Baines doubled the Toffees' lead from the penalty spot.
Cheltenham skipper Alan Bennett was adjudged to have fouled Fellaini and the resulting spot-kick from Baines contained too much power for Brown even though he dived the right way.
Fellaini got into another good position inside the six yard box but glanced his header wide from Coleman's centre.
The impressive Osman appeared to have killed off the tie after 49 minutes when he slotted the ball past Brown after taking a Baines pass in his stride.
Cheltenham pulled one back two minutes later through Penn's cool finish after good play by Mohamed.
But Coleman re-established Everton's three goal lead after 57 minutes. He burst into the box and dinked the ball over Brown after exchanging passes with Victor Anichebe.
It needed a fine save from Brown to deny Naismith before Fellaini made it 5-1 with a minute left from Victor Anichebe's cross.
|Cheltenham Town||Team Statistics||Everton|
|0||1st Half Goals||2|
|3||Shots on Target||10|
|8||Shots off Target||6|
|45.3||1st Half Poss.||54.7|
|37.7||2nd Half Poss.||62.3|