The Cottagers have had to pay an initial £25,000 which could rise to £250,000 with add ons, plus guarantee two home friendlies with the Gander Green Lane outfit.
The 18 year old will stay with Sutton on loan until January.
“Stefan has got half a chance to make it as a professional. He has raw pace, strength and the right attitude. We’re absolutely delighted for him,” said boss Paul Doswell.
“If he doesn’t make it at Fulham, with the schooling he’ll get there he’ll get another chance lower down the pro level.
“He will be a loss, but in terms of pure talent Billy Dunn is as good, if not better. He is the best I’ve ever seen and could be the next one to push on.
“I’ve been critical in the past, but Stefan’s example - in particular, the way he has gone about his business - can show him what he could achieve.”