There were also victories for Leicester City, who needed extra time to beat West Ham United 2-1, Aston Villa, who won 1-0 at home to local rivals Birmingham City, and Everton who came from behind to beat Reading 2-1.
Swansea City suffered a surprise 1-0 defeat at Hull City, Stoke City booked a spot in the next round by winning 1-0 at second-tier Fulham and top flight Bournemouth came through a penalty shootout with Preston North End.
There was never even a hint of an upset in the offing at Sunderland, whose lack of confidence after a poor start to the season was ruthlessly exposed by a rampant City.
Black Cats coach Advocaat was feted as a hero after steering the north-east club to safety last season, but his star has waned at the start of the current campaign as his side are without a win in six Premier League games.
They fell behind after nine minutes to a Sergio Aguero penalty after Patrick van Aanholt clattered into Jesus Navas and the Argentine forward dispatched the spot kick with a deft chip.
The floodgates then opened as three goals in 11 minutes from Kevin De Bruyne, an own goal by Sunderland keeper Vito Mannone and a Raheem Sterling strike put City 4-0 ahead at halftime, before Ola Toivonen headed a consolation in the second half.
Swansea were the only Premier League team to exit at the hands of lower league opposition on Tuesday as Hull's David Meyler struck four minutes before halftime for the side relegated from the top flight last season.
The midfielder stretched for a rebound and swept the ball into the corner four minutes before halftime to seal the tie.
