The 24-year-old Scotland international has signed for an undisclosed fee, understood to be in the region of B£2.5million.
Bridcutt, who will wear the number 23 shirt, had not played for the south-coast club since submitting a transfer request earlier this month after the Black Cats formalised their interest in him.
Poyet had targeted Bridcutt and team-mate Will Buckley since the beginning of the month, with the club's interest in the pair perhaps signalling that director of football Roberto De Fanti's reign was drawing to a close.
The Italian, who was relieved of his duties shortly after the bids were tabled, oversaw the acquisition of 14 players during the summer, 13 of them from overseas.
Poyet worked with the pair during his spell in charge at the Amex Stadium and his admiration for Reading-born Bridcutt in particular is well-known.
The former Chelsea trainee had loan spells with Yeovil, Watford and Stockport before joining the Seagulls, where he made more than 150 senior appearances in 2010.
He was handed a senior debut for his country against Serbia in March last year.
Bridcutt becomes the Black Cats' fifth signing this month with Spaniard Marcos Alonso and Argentinians Oscar Ustari, Santiago Vergini and Ignacio Scocco having already been drafted in despite De Fanti's departure.