Norwich went into Saturday's third-round tie at London Road on the back of four successive losses in the league over Christmas and New Year but eased to a comfortable 3-0 win despite Hughton making eight changes.
The Canaries resume their league campaign on Saturday with a home match against out-of-form Newcastle, while they now have a fourth-round tie to look forward to later this month.
Hughton said: "Although they've been very tight games, we were on the back of four defeats so, irrespective of what the next game was, it's one you want to win.
"We want to stay in the cup for as long as possible and I think you can gain confidence when you are able to use the squad and see good attitudes."
Of the team that started last Tuesday's defeat by West Ham, only Russell Martin, Ryan Bennett and Robert Snodgrass kept their places.
Snodgrass was the standout player but Elliott Bennett and Simeon Jackson both took a rare chance to impress up front with a goal apiece, while Bennett created the third goal for Snodgrass.
Hughton said: "Considering both of them haven't played as much football as they would like, I thought they were good right through the 90 minutes. They were a constant threat and gave us really good options up front."
Hughton has been reluctant to rotate his Premier League team and he indicated the players who took their chance on Saturday may have to wait for another.
"You never want to say no," he said when asked if he had been given a selection dilemma.
"We've had a fairly settled side but that's why the players train like they do and that's why you want the competition.
"When you look at players like (Marc) Tierney and (David) Fox, Elliott Bennett and Simeon, that haven't had the football, you wouldn't have seen that because I thought they carried themselves very well, looked very fit and had a lot of energy in their game.
"Of course the result is what you want but it's the attitude and the performance that pleased me the most.
"I think whatever the game was, I would have played the same team. It's about timing and we've also got a team that have played a lot of games together. Those players deserved to play, and they've put themselves very much in the frame for the next round."
Peterborough boss Darren Ferguson was furious with his side's display in front of their biggest crowd of the season and left his players stewing in the dressing room for an hour before talking to them.
The result was even more disappointing because it came after their best run of form this season, with four wins from five Championship games helping them climb out of the relegation zone.
Ferguson is hoping to bring in reinforcements this month, while six players are on the transfer list, but he is keen to avoid knee-jerk reactions on the back of Saturday.
He said: "We've tried this week to get some in but we haven't managed to do it. January's a really important month in terms of people going out and people coming in.
"If you ask me straight after the game I've just watched, I would say I need players in. But I'll assess it in more depth when I watch the game again."