The Ashington-born striker, who was wearing Newcastle's famous number nine shirt, scored the opening three goals, with Adam Armstrong adding a late fourth in front of more than 3,500 spectators.
Newcastle's juniors were dominant throughout, with Sunderland failing to trouble goalkeeper Freddie Woodman as they produced a surprisingly ineffective display.
The hosts claimed the lead a minute before the interval with Rolando Aarons surging into the box before cutting the ball back for Quinn to score.
Quinn added a second when Sunderland goalkeeper Maksymilian Stryjek failed to hold Greg Olley's cross at the start of the second half, and the England under-17 international completed his hat-trick when he converted Jamie Sterry's centre.
Armstrong capitalised on a poor back-pass for claim Newcastle's fourth goal and set up a fifth-round tie with either Stoke City or Derby County.
Newcastle: Woodman, Sterry, McKinnon, Satka, Gillesphey (Drennan), Gilliead (Heardman), Olley, Kemen, Aarons, Armstrong, Quinn (Barlaser).
Sunderland: Stryjek, Ledger, T Robson, E Robson (Wright), Colquhoun, Bealding, Robson (Lowrie), Smith, Stinson (Nelson), Gooch, Greenwood.