The 17-year-old, who was born in Brighton, has been on trial with Albion this summer and will initially join the club's Under-21 squad.
Head of football operations David Burke told Brighton's website: "We are really pleased to have signed Jesse, not only because he has come through the ranks at Chelsea and because of the training he had there, but also because he is a local lad.
"Brighton was a centre of excellence when he chose to go to an academy with Chelsea but now we are identifying local talent and attempting to bring them back to our academy.
"Jesse is one of the players who falls under that umbrella, because he is a local boy and we are delighted he has joined the club.
"It is a mark of how far this club have come, with the new stadium, the new training ground on the horizon, the academy up and running, so we can now attract players of the calibre of Jesse."