The Potters needed penalties to progress after the match had finished 1-1, with substitute Geoff Cameron scoring the winning spot-kick after home defender Scott Griffiths had put his effort against the crossbar.
Hughes, who will now take his side to one of his former clubs in Championship outfit Fulham for round three, said: “I think the the obvious feeling is relief. We had opportunities to win the game, we were 1-0 up with 90 minutes gone, about 40 seconds left and we allowed them the one chance that they needed.
“They deserve credit because they kept going and were competitive. They had one chance and put it away, you’re thinking at 1-1 we’re making hard work of this.
“But thankfully we came through extra time without ever really, on the night, creating enough chances for the play that we had,
“In the end we’re grateful to go through on penalties and that was the order of the day, to progress in the competition.
“The lads stepped up and showed good determination to convert the penalties and we went through in the end.”
Stoke had taken the lead through Jonathan Walters’ instinctive finish as he collected Stephen Ireland’s dinked through ball to lift his effort over the onrushing Elliot Justham.
Luton then forced extra-time thanks to Cameron McGeehan’s stoppage-time leveller with the substitute arriving at the far post to convert Griffiths’ low cross.
Hatters boss John Still was proud his team had matched up with a side who finished ninth in the Premier League last term and chose a starting 11 packed full of internationals.
“You’re always disappointed when you don’t win, but 120 minutes against a Premier League side was absolutely terrific,” he said.
“I thought everybody was at the level that you would want. We had some good individual performances, a good collective performance.
“We haven’t done well in the league but the players showed they are players who are more than capable.
“It was a clever run from Walters for the goal and that was the only real time we felt in danger.
“Our goal was a classic goal and we threatened a few times, we didn’t just defend, defend, defend – we moved out quickly.
“It was a really good evening for us. I was really pleased with the performance but it’s always been never too high, never too low.”