After a goalless first half the young Glovers scored three second half goals to defeat the side from Gloucestershire. Town goals came from Jack Ball, fellow striker Paul Akinuli, Samyan Ismail and Ollie Bassett.
In the first half the visitors had goalkeeper Luke Merchant to thank after making two good saves to deny the young Glovers a deserved lead.
The best chance of the half fell to Glovers winger Strobel, but his effort was well palmed away by the shot stopper.
At the other end the centre back partnership of Sam Boulton and captain Jayden Corbridge kept the visitors at bay whilst looking calm and composed.
Town started the second half on the front foot and opened the scoring with 54 minutes on the clock.
A neat one-two between the impressive Corbridge and Ball saw the Ball get away from his markwer in the area to fire a shot across Merchant and into the bottom corner of the net.
Town doubled their lead on the hour mark when striker Akinuli headed home at the back post after Town recycled a half cleared corner well.
The Glovers added a third on 68 minutes and what a goal it was. A superb chip pass from Ollie Bassett just outside the area released Ismail, who made no mistake in hammering home from 12 yards.
The impressive Bassett made it 4-0 when he picked the ball up on the edge of the Rovers area and hit an unstoppable shot into the far corner, leaving Merchant with no chance.
Town now face a home time in round two against Met Police Under 18’s at Huish Park.
Yeovil Town: 1. James Needles, 2. Ren Griffiths, 3. Tiago Maquis-Sa, 4. Max Melanson, 5. Jayden Corbridge, 6. Sam Boulton, 7. Samyan Ismail, 8. Jack Ball, 9. Paul Akinuli, 10. Ollie Bassett, 11. Philipp Strobel. Sub: 12. Kingsley Latham, 17. Prince Iwyen-Nwabudike, 14. Arthur Warren, 15. Taylor Scraff, 16. Dion Macfoy-Johnston.