The 27-year-old has signed a two-and-a-half-year-deal with the Cottagers who has already linked up with Scott Parker's side. There is an option to extend the deal with a further 12 months.
Hector has been training with Fulham, and will continue to until his deal is confirmed in January.
Talking to fulhamfc.com, Hector shared his delight at making a permanent move down the road from Stamford Bridge: "It was an easy decision to make. My Mum’s from Fulham, so it’s nice to join a club that some of my family support.
"It’s good to now be a part of the setup here. Hopefully things can work out well and we can get promoted. I know how tough the Championship is and it won’t be any different for us.
"We’ll have to work hard for it but the squad is very good and my job will be to help them push for promotion."