The 24-year-old midfielder, who was born on Tyneside and supported Newcastle as a boy, signed a long-term contract after rejecting a new deal at the Stadium of Light.
Colback, who had been with the Wearside club since he was eight, said: "To come to the team I supported as a boy will be really special for me. It was an opportunity I couldn't let slip."
Sunderland, however, said in a terse statement: "Jack is a player we have nurtured and developed through our academy system since he was eight years old. We gave him the opportunity to become a professional footballer and are therefore extremely disappointed in the events that have led to his departure."