Following a goalless first period, Samyan Ismail set the young Glovers on their way just after the interval.
Paulius Golubickas added a second for the hosts before Jeremiah Gyebi put extra gloss on the victory late on.
Town showed early signs of progress, looking to exploit Cheltenham in behind but the visitors looked dangerous on the counter, sending two warning shots wide inside five minutes.
Home goalkeeper Chandler Hallett was a late call-up to the starting line-up after Tommy Scott was injured in the warm-up, and he needed to be at his best to tip Adam Hockey’s sweet volley round the post.
At the other end, Robins stopper Ross Grimshaw kept the score at 0-0 with two fine stops from Neville Nzembela efforts.
An intricate passage of play from Yeovil ended with Nzembela one-on-one and curling shot was pushed over by Grimshaw.
In the second head-to-head between the two, a similar outcome occurred. Town youngster Nzembela attempted a low strike towards the bottom corner by Grimshaw was equal to it once again.
The Under-18s continued their dominance in the second half and opened the scoring on 50 minutes.
Ismail picked up the ball 25 yards from goal, skipped past three Cheltenham defenders and – with a hand of the post – found the back of the net from an acute angle.
One became two shortly afterwards. A mazy dribble from Golubickas was rewarded as he fired high into the roof of the goal.
Untroubled and a spectator for the most part, Hallett made an excellent save to deny George Lloyd before Gyebi rounded off the scoring, collecting Bradley Cherrett’s slide-rule pass and hitting in off the woodwork.
Yeovil Town U18: Hallett, Fripp, Gyebi, Cherrett (Haughton 90), Dix (Hayes 69), Rice-Lethaby (Arnold 64), Wan, Hilton-Jones, Ismail, Golubickas, Nzembela (Rogers 64).