Goals from Jelavic, Baines, Osman, Coleman & Fellaini saw the Premier League side ease past the League Two outfit, who grabbed a consolation through Russell Penn
Nikica Jelavic, Leighton Baines, Leon Osman, Seamus Coleman and Marouane Fellaini all found the back of the net against the League Two outfit, with Russell Penn's second-half strike earning some consolation for a spirited, if easily beaten, Robins side.
Despite making four changes to the team that saw off Newcastle 2-1 last week, David Moyes named a strong Everton team for the trip to the Abbey Business Stadium, with Marouane Fellaini and Baines both starting and striking duo Victor Anichebe and Jelavic came into the XI.
Marlon Pack and Kaid Mohamed were both handed starts by manager Mark Yates, who made two changes after his side fell to a 1-0 defeat away to Oxford United on New Years' Day.
The visitors were quick to make their superiority on paper count, with Jelavic pouncing to hand the Toffees an early lead. After the ball bobbled loose in the box, Fellaini took the chance to shoot and hit the post, with the Croatian quickest to react, stretching to poke home the rebound.
The hosts had started brightly, although Darren Carter failed to truly test goalkeeper Tim Howard with a number of early chances. Cheltenham lived to regret their lack of proficiency in front of goal.
Captain Alan Bennett clumsily fouled Fellaini in the box, with his robust marking of the Belgian adjudged to have been worthy of penalty by referee Kevin Friend, and up stepped Baines to double the lead and put the Premier League side in firm control of the game.
The hosts rallied to try and provide some hope to the increasingly deflated crowd going into half-time, but Howard was never forced into making a save, while Everton's Premier League class allowed them to steamroll forward almost at will.
And such a pattern continued into the second 45. Baines was allowed to drive down the left-hand side, leaving the Cheltenham back line to hopelessly appeal for the offside flag. In the meantime, the Everton captain picked out Osman, who coolly skipped past the onrushing centre-back before slotting home to make it 3-0.
However, just as Everton looked to be on course to rack up an embarrassing scoreline, Mohamed carved out an opening for Penn, who finished smartly.
The respite was short lived, with a cute effort from Coleman restoring Everton's three-goal advantage just seven minutes later.
Job done and progression sealed, Moyes opted to rest Baines and Osman, with Magaye Gueye and Steven Naismith handed run outs as the game drew to a close. The only thing missing? A goal for Fellaini. And after a dominant midfield display, the Belgian was sure to get his name on the scoresheet, guiding home from an Anichebe cross on the stroke of 90 minutes.
A trip to Bolton or Sunderland awaits at the end of the month, while the visit of Swansea City to Goodison Park in the Premier League is next up for the Toffees.