The Seagulls will pay a fixed £46 million for the 19-year-old centre-back, with £4 million in performance-related add-ons.
Vuskovic, who featured for Croatia at the World Cup, has penned a five-year contract until 2031.
Spurs have also secured a 20% sell-on clause on any profit made in a potential future sale.
Vuskovic signed for Tottenham from Hajduk Split in 2023 as a 16-year-old, costing around £12 million.
He officially joined the club in July 2025 and scored during a pre-season friendly against Reading.
The 19-year-old was loaned to Hamburg in search of regular first-team minutes and shone during his Bundesliga stint.
He was named in the Bundesliga Team of the Season and won the Goal of the Year award for an improvised scorpion kick.
Vuskovic has six caps for Croatia and became their youngest-ever World Cup player in the summer.