Boss Mauricio Pochettino made nine changes for the Cup clash from the side that stunned Chelsea last Wednesday, but Ben Davies' header and a late strike from Son Heung-min were enough to seal Spurs' progress at White Hart Lane.
Dier and Toby Alderweireld were the only players to retain their starting places, with Dier handed the armband in the absence of regular skipper Hugo Lloris and the Frenchman's stand-in Harry Kane.
Dier has been earmarked as a potential Spurs captain of the future for his confident personality and commanding style of play, and it is a responsibility the 22-year-old would relish.
"Would I like to be club captain? I hope so, maybe one day," Dier said. "We have some fantastic captains here at the moment but I'm still young.
"Hopefully one day maybe I will get that honour but I am focused on playing for Tottenham, nothing else.
"I like to think leadership comes naturally. It doesn't affect me. When I was younger, going through age groups at Sporting (Lisbon), I was captain and I was captain before - it's not something new to me.
"But I'm not the man to tell you if I'm a good captain or not. Other people are."