A record fee, understood to be £13.5m, was agreed with the Belgian club for the 25-year-old last week, only for work permit issues to hold up the deal.
Those issues have now been sorted, and the Colombia international will be free to make his debut against Barnet in Tuesday's EFL Cup tie, provided international clearance is received in time.
Seagulls boss Chris Hughton told reporters: "This has been a very complex deal, but I am absolutely delighted we are finally in a position to confirm it is completed. I would like to thank all those involved for their hard work and am delighted Jose is now able to begin work.
"He has already been in Brighton & Hove for a period of time, and been able to meet his new colleagues, but has not been permitted to train with us - so he is now very keen to get started, and we are looking forward to working with him on the training pitch.
"He's a very exciting player to watch, and along with that excitement he will bring pace and creativity to the team; he also has an excellent goalscoring record."
Brighton broke their transfer record last Monday when they signed midfielder Davy Propper from PSV Eindhoven for around £6m, and prior to that, had paid a reported £5m in June for Valencia goalkeeper Mathew Ryan.