The Blues accepted a bid in the region of £9million on Thursday despite concerns over their own midfield, with Conor Gallagher out of form and N’Golo Kante set for a spell on the sidelines.
A £40m approach for Edson Alvarez ahead of transfer deadline day failed to reap any results as Chelsea also missed out on Dynamo Moscow youngster Arsen Zakharyan.
Brighton’s bid for Gilmour was accepted nonetheless, with the player’s move to Brighton confirmed well beyond Thursday night’s deadline.
The 21-year-old has signed a four-year contract with the Seagulls, and could make his debut this weekend against Leicester.
“Billy arrives with a fantastic pedigree, having played in the Premier League and Champions League as well as for Scotland at last year’s Euros,” Brighton boss Graham Potter said after the move was officially confirmed.
“He will complement our existing midfield options. That experience at quite a young age shows the strength of character he has and we’re really excited by his potential.
“Now it’s about allowing him the time to get used to a new environment and settling in to our club. We can’t wait to work with him.”