League Cup Wrap: Black Cats progress
Caretaker boss Kevin Ball saw Emanuele Giaccherini put the home side ahead in the 32nd minute before Valentin Roberge came off the bench in the second half to double their advantage 15 minutes from time.
Tottenham and Manchester City enjoyed comfortable passages into the next round with victories over Aston Villa and Wigan Athletic respectively.
Jermain Defoe scored at the end of each half, either side of goals from Paulinho and Nacer Chadli, as Spurs swept aside Villa 4-0 at Villa Park.
City went one better in seeing off Championship outfit Wigan 5-0 at the Etihad Stadium.
Edin Dzeko broke the deadlock in the 33rd minute before Stevan Jovetic and Yaya Toure made it 3-0 with second-half strikes.
Jovetic bagged his second with seven minutes to go before Jesus Navas scored his first goal for his new club four minutes from the end to round off the scoring.
Chelsea, like City, also overcame lower-league opposition as they defeated Swindon Town 2-0 at the County Ground.
Fernando Torres put them ahead in the 29th minute before Ramires' strike six minutes later gave Chelsea the victory.
West Ham and Fulham were victorious in the all-Premier League affairs.
Fulham came from behind to beat Everton 2-1 at Craven Cottage, as goals from Dimitar Berbatov in the 54th minute and Darren Bent in the 68th proved good enough to overcome Steven Naismith's 12th-minute opener.
West Ham got off to a flying start in their 3-2 win over Cardiff City at Upton Park, goals from Ravel Morrison and Matt Jarvis giving them a two-goal lead inside eight minutes.
The visitors hit back through Craig Noone on the stroke of half-time, before Peter Odemwingie drew them level with 15 minutes to go.
The comeback was in vain, however, as Ricardo Vaz Te scored with two minutes remaining to send Sam Allardyce's side through to the next round.
Norwich City prevailed in the only tie to go to extra time, coming from two goals down to win 3-2 away at Championship Watford.
Javier Acuna put the home side in front midway through the first half, before Marco Faraoni doubled the lead 10 minutes after half-time.
Substitute Josh Murphy got one back for Norwich with 13 minutes remaining, before Gary Hooper scored his first goal for the club in the fourth minute of stoppage time.
And it was Hooper who won the day for his new club, netting five minutes before the end of extra time to send Norwich into the fourth round.
Southampton were another Premier League side to win through, thanks to a 2-0 home defeat of League One strugglers Bristol City.
Gaston Ramirez put his side ahead in the 15th minute before Jos Hooiveld's header seven minutes from time sealed a comfortable victory.
Hull City will join them in the hat for the fourth round after Nick Proschwitz's goal after 59 minutes saw them defeat Huddersfield Town 1-0, while there was a 2-1 win for Burnley over fellow Championship side Nottingham Forest.
Matt Derbyshire put Forest ahead in the 24th minute, but Danny Ings equalised on the stroke of half-time before sealing the win with his second midway through the second half.
Elsewhere, Leicester City also came from behind as they beat Derby County 2-1 at home.
Chris Martin put Derby in front three minutes before the break but Anthony Knockaert equalised 12 minutes from time before Danny Drinkwater found a winner three minutes later to send Nigel Pearson's side through.