Seamus Coleman scored the goal of the night as Everton reaped the rewards of fielding a virtual full-strength side to cruise into the FA Cup fourth round.
David Moyes' side will now visit the winners of next week's Sunderland v Bolton replay at the Stadium Of Light in the fourth round.
Moyes paid Cheltenham the compliment of parading eight of the line-up that started the 2-1 mid-week win at Newcastle in the league.
Everton's decisive start to the game, coupled with the powerful nature of the team, ensured there was little chance of a shock result.
Fellaini was a constant threat in a free role while full-backs Coleman and Baines got forward to good effect.
But promotion-chasing Cheltenham showed plenty of fight with Penn and Darren Carter impressing in midfield.
Coleman's fine goal, Everton's fourth of the night, re-establishing a three-goal lead, capped off an impressive performance from the young Irishman.
He burst into the box and dinked the ball over Brown after exchanging passes with Victor Anichebe.