Early strikes from Nikica Jelavic and Leighton Baines shelved any local hopes of a shock result and effectively ended the contest before halftime.
Leon Osman, Seamus Coleman and Marouane Fellaini then added to the scoreline after the break, while Russell Penn grabbed Cheltenham's lone riposte.
Here are six Everton-related talking points to emerge from the game.
Prior to kick off, there were several phantom Everton XIs doing the rounds on Twitter, although few predicted the strength of David Moyes' eventual lineup.
Instead of the presumed presence of youth and fringe players, the Toffees went with one of their strongest available teams, with Steven Pienaar the only fit first-team regular to miss out.
As important as it is to keep pace with the Premier League's top four, this selection illustrates just how fiercely Moyes covets a trophy.
Marking his Everton tenure with silverware is clearly high among his priorities.
Seamus Coleman returned from injury to produce another accomplished attacking display from right-back.
The Irishman scored a memorable goal by deftly lobbing the goalkeeper and was a constant threat going forward throughout the game.
Certain to make the odd defensive gaffe as he hones his craft at the back, Coleman's attacking impetus is already becoming a vital facet of his side's play.
His direct running and end product have improved dramatically, and with opponents preparing to nullify Leighton Baines and Everton's left, his right-sided input comes as a useful distraction. Especially when accompanied by a fit-again Kevin Mirallas.
His return also enables Phil Jagielka to revert to centre-back, instead of being wasted as a makeshift right-back.
A huge positive from tonight's result was Nikica Jelavic's return to scoring ways.
Despite his impressive assist against Newcastle, the Croatian forward rarely offers anything substantial in the creative stakes, instead preferring to seek out the final ball.
Therefore when he's not at his most potent, lethal best, his side often struggle to finish off opponents.
Regardless of the level of opposition, finding the net after almost a month without a goal should do wonders for his confidence, invigorating him for Saturday's Premier League clash with Swansea.
Whether or not Jelavic finds form in the coming months may prove decisive in Everton's quest for Champions League football.
As favourably as his starting lineup was received by supporters, Moyes' choice of substitutions was a little more controversial.
Few had an issue with Steven Naismith's emergence from the bench, but opting to use Thomas Hitzlsperger and Magaye Gueye—instead of Ross Barkley—were decisions especially hard to decipher.
Hitzlsperger is on a short-term contract and isn't likely to remain long at Everton, while Gueye has rarely impressed despite several cameos from the bench.
Surely giving the stage to Barkley and allowing him the license to express himself and gain confidence would have been a far better call for Everton's immediate and long-term future.
If nothing else, this omission only cements the fact the youngster must once again be loaned.
After a typically commanding hour on the field, in which he notched up his fourth consecutive FA Cup third round goal, Leighton Baines was hauled off for a well-deserved rest.
At least that's what fans thought. However, as soon as he left the field he was greeted with a pair of icepacks which were immediately placed around his ankles.
Moyes was quizzed on the subject post match, saying:
Leighton has got a bad ankle problem which I need to watch and be careful with that. I felt we needed him because we have to respect the players at this level. (via BBC Sport)
Certainly stronger words than many expected.
While the injury does not seem to be a hindrance to his efforts on the pitch, it is clearly an issue and something Moyes needs to monitor closely.
Maintaining a Champions League challenge without Baines would be a particularly tough ask for this side.
The only obvious negative to detract from this game was Everton's inability to finally register a clean sheet—especially facing such lowly opposition.
Will they ever shut out an attacking unit again?
Despite starting with arguably their first choice back four, the Toffees still appeared unconvincing at set plays and the fact that even a League Two outfit could penetrate them is bound to concern Moyes.
This run now stretches an unbelievable 18 games in all competitions, dating back to a 3-0 win over Swansea at the end of September—coincidentally the side Everton face next.