Goals from Jelavic, Baines, Osman, Coleman and 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 side for the trip to the Abbey Business Stadium, with Fellaini and Baines both starting while striking duo Victor Anichebe and Jelavic came into the XI.
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 American goalkeeper Tim Howard with a number of early chances. Cheltenham would regret the 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 Baines stepped up to double the lead from the spot 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 halftime, yet Howard was never forced into making a save, while Everton's Premier League class allowed the Toffees to steamroll forward almost at will.
And such a pattern continued into the second half. 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 on-rushing center back before slotting home to make it 3-0.
Just as Everton looked to be on course to racking up an embarrassing scoreline, Mohamed carved out an opening for Penn, who finished smartly.
The respite was short lived, though, with an effort from Coleman restoring Everton's three-goal advantage just seven minutes later.
Fellaini, after a dominant midfield display, was sure to get his name on the scoresheet as well, 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 league is next up for the Toffees.