Norwich eased to a comfortable 3-0 win on Saturday, on the back of four successive losses in the league over the festive period.
Hughton's men resume their Premier League campaign next weekend with a home match against out-of-form Newcastle, along with a fourth round tie to look forward to.
"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," said Hughton.
"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."
The Norwich manager made eight changes to his side ahead of the FA Cup clash, with Russell Martin, Ryan Bennett and Robert Snodgrass the only names that remained in the line-up from Tuesday's defeat to West Ham.
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."
Snodgrass stood out, but Elliott Bennett and Simeon Jackson both took a rare chance to impress up front with a goal each, while Bennett created the third goal for Snodgrass.
"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," he added.
"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 disappointed with his side's display in front of their biggest crowd this season.
Unlike Norwich, the result came after their best run of form this term, with four wins from five Championship games helping them climb out of the relegation zone.
Hoping to bring in some reinforcements this month, Ferguson 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."