With the two teams feeling each other out in the early stages, Marouane Fellaini had the first crack at goal, firing a snapshot wide from 25 yards.
Carter had the home side's first chance from a very promising opening but Tim Howard was grateful to see him scuff his effort wide.
Bryan Oviedo, starting in place of Steven Pienaar then had a weak shot from the edge of the box but the Costa Rican was involved almost immediately afterwards for the opening goal.
His backheel on the 18 yard line found Fellaini and though the Belgian curled an elegant shot off the face of the post, Nikica Jelavic was on hand to turn the rebound home.
The Croatian might have doubled his tally three minutes later but half-volleyed Fellaini's knockdown wide from 12 yards.
Far from over-awed, though, the League Two side tested Howard for the first time with 17 minutes on the clock but the American got down low to make a comfortable save from Carter.
A foul on Anichebe in Leighton Baines territory a minute later offered the left-back a chance to repeat his heroics on Tyneside last week but the 'keeper palmed away his more finessed effort.
The ball stayed alive for Everton, though, and when Bennett climbed all over Fellaini in the box, referee Kevin Friend pointed to the spot. Baines stepped up to take the penalty and coolly slotted it into the corner.
Everton had a grip on the tie now but their patient probing didn't yield another shot at goal until the half-hour mark when Fellaini swept one a couple of yards wide.
Cheltenham, meanwhile, posed their biggest threat from set-pieces but back-to-back corners in the 33rd minute yielded nothing but a foul on Howard as he tried to punch the ball clear from his six-yard line.
And when Baines tripped McGlashan as he tried to power his way in from the right flank, Jelavic had to head clear the ensuing free kick from Jones. Panic in the Everton area from the resulting throw-in set up a chance for McGlashan but he side-footed over the bar.
Back at the other end, Fellaini glanced a stooping header wide of the far post from Coleman's cross from the right five minutes before the break.
And Baines proved too quick for McGlashan and the Cheltenham winger scythed him down but referee Friend kept his cards in his pocket for what would ordinarily be a yellow card offence. The Blues' left back's free kick from the left touchline was too low, though, and was headed away to safety from the home side's point of view.
The second period began in familiar fashion to how the first had ended and Anichebe sliced the first shot of the half well wide after space opened up in front of him 25 yards from goal.
Osman, though, lit up the evening with a lovely goal, taking Baines' cut-back before jinking calmly around a defender and tucking it under the 'keeper to make it 3 - 0.
Howard's hopes of keeping a clean sheet were dashed within only a couple of minutes, however, when Penn was found on the left side of the Everton area and he smashed it past the American 'keeper to make it 3 - 1 with just five minutes played in the second half.
Everton resumed their offensive and Osman saw a deflected half-volley easily caught by Brown but the 'keeper could do nothing to prevent number four.
Anichebe clipped the ball in from the right wing to meet Coleman's run and the Irishman dinked it over Brown with a deft touch with 57 minutes gone.
Moyes felt comfortable enough with a little over an hour gone to make his first change, removing Baines and putting Magaye Gueye on the left wing and dropping Oviedo back to left back. Steven Naismith followed from the bench five minutes later replacing Osman.
Cheltenham's hoped to make a game of it and substitute Taylor smashed a half-volley high and wide, Bennett steered a towering header past the far post from a tempting cross and Penn warmed Howard's palms with a 25-yard drive.
Hopes that Ross Barkley would get an outing were left unfulfilled 10 minutes from time when Thomas Hitzlsperger replaced Jelavic as the last Everton change of the night.
Neither Gueye nor Naismith did much to press their case for a permanent place in the starting XI but the Scot did pop up in front of goal a few minutes from time to latch into Coleman's cut-back but he could only shoot straight at Brown.
The fifth Everton goal did arrive in the last minute of the 90, though, when Anichebe raced down the left channel and centred perfectly for Fellaini to side-foot home with aplomb.
Everton embark on what will hopefully be a long trip along the road to Wembley as they face League Two Cheltenham Town in a Monday-evening kick off at Whaddon Road.
Mark Yates' side earned the right to face the Blues when they overcome Hereford United in an FA Cup Second Round replay and their attempts to spring an act of giant-killing on last season's semi-finalists will be broadcast live on ESPN.
Mindful of the prior cup upsets that he has experienced against Shrewsbury Town and Oldham Athletic during his time at Goodison, David Moyes used his pre-match press conference to stress that his side will be taking nothing for granted against the Robins.
The Blues' manager is expected to make changes to a team that was tested physically over a busy festive season and showed signs of fatigue as they stood firm at St James' Park last week to hold onto all three points against Newcastle in the Premier League, but will likely not go to the same extent as he did at Leeds in the League Cup earlier this season.
In that fateful tie, Moyes made six changes and fielded an experienced midfield that ended up succumbing to the Yorkshire side 2-1 but he is expected to retain the core of his first-choice team for this one while perhaps resting a couple of the ever-presents.
Having spent three matches on the sidelines through suspension before the win at Newcastle, Marouane Fellaini is almost guaranteed to play and Victor Anichebe could get the nod to start up front in place of Nikica Jelavic who has looked in need of some time to recharge his batteries in recent games.
Leon Osman as run himself ragged for the cause and both Steven Pienaar and Leighton Baines played against the Magpies despite being rated as doubts before that game so one or two of that trio could also be handed a rest. Phil Neville, meanwhile, could be omitted after receiving eight stitches in a head would last Wednesday.
The likes of Thomas Hitzlsperger, Ross Barkley and Bryan Oviedo are all be waiting in the wings to step in if asked while Seamus Coleman could be fit enough to start at right back. Kevin Mirallas is said to be making good progress again as he recovers from a hamstring injury but is not expected to be fit, while Tony Hibbert and Darron Gibson are also definite absentees.
With the draw for the Fourth Round already made, Everton know that they would end up with another away game at either Bolton or Sunderland in the next phase should they safely negotatiate a Cheltenham side who are in playoff contention near the top of League Two.
Despite the lift that the occasion and home field advantage will provide the Robins, this should prodive Everton with a fairly simple route through. It's unlikely to be a comfortable ride but, as the top-flight team that's only been beaten three times all season, the onus is on them to get the job done.
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