Marc Muniesa's double was enough to seal a 2-1 win at the Stadium of Light as the visitors came from behind to progress.
Hughes revealed that success in the competition is one of Stoke's top priorities this season as they look to emulate the club's only previous triumph in 1971-72.
Stoke reached the quarter-finals last season before being knocked out by Manchester United and the Welshman hopes to build on Tuesday's win to go a couple of steps further.
"It has been something we have talked about prior to the season starting and we are making good progress," he said.
"We are only in three competitions and realistically we can only win two of them, so we wanted to try to do our best in this one.
"It was not an easy draw for us, being away at a Premier League ground, so to come here and get through to the next round is important given the draw last time out, a lot of Premier League teams coming up against each other.
"So we are grateful that we are in the hat because a number of Premier League teams will go out before the draw is made.
"Everybody knows that you need a little bit of luck in cup competitions, that's a given, but if we can approach games and take games away from good teams then we have got a chance."
Stoke will learn the identity of their opponents in the fourth round when the draw takes place on Wednesday.
