Harry Wilson's early strike and second-half goals from substitute Andreas Pereira and Layvin Kurzawa earned the Londoners a 3-2 fourth-round replay victory at the Stadium of Light, when their boss was forced to take a seat among a crowd of 29,651 as he served a touchline ban.
Their reward is a home clash with Leeds, who parted company with head coach Jesse Marsch on Monday.
For Championship Sunderland, who dragged themselves back into it at 2-1 courtesy of Jack Clarke's curled effort and 3-2 through substitute Jewison Bennette, the dream of a reunion with the club they famously beat in the 1973 final evaporated despite a concerted fightback.
A youthful line-up - nine of Tony Mowbray's starting XI were aged 23 or under with 15-year-old Chris Rigg once again among the substitutes - fell behind with eight minutes gone when Carlos Vinicius did well to hold off his marker and lay the ball back to Wilson, who stabbed into the bottom corner with the outside of his left foot.
The combination of Tom Cairney, Wilson and the dangerous Manor Solomon repeatedly stretched the Black Cats, who struggled to break out of their own half despite the best efforts of the lively Amad Diallo and Abdoullah Ba.
Wilson might have doubled his tally with 25 minutes gone after picking off Patrick Roberts' ill-judged cross-field pass to Aji Alese and racing in on goal, although goalkeeper Anthony Patterson easily claimed his weak attempt, and the Wales international narrowly missed the target after being played in by Cairney minutes later.
The home side forced their way into the game in the latter stages of the half with Clarke seeing a deflected effort hacked away before Amad's dipping thunderbolt caused goalkeeper Marek Rodak all kinds of problems.
Sunderland returned after the break still very much in the tie, although their task was not made any easier by Silva's introduction of Aleksandar Mitrovic and Pereira before play resumed.
Pereira almost made a swift impact when he broke from deep inside his own half and combined with Solomon and Wilson to set up Cairney to fire just wide.
The Black Cats replied in kind, Dan Neil curling just past the post from the edge of the box before Rodak had to turn away Roberts' strike, and it took a goal-line clearance by Shane Duffy to deny Amad a 57th-minute equaliser.
However, they fell further behind within two minutes when former Newcastle striker Mitrovic pulled the ball back for Pereira to fire home from close range.
Clarke gave the hosts hope when he dispatched a curling 77th-minute shot into Rodak's top corner, but Kurzawa restored the visitors' two-goal advantage within five minutes and although substitute Bennette reduced the deficit once again as the game entered stoppage time, there was no way back.