Kenwyne Jones restored Stoke's lead four minutes into the additional 30, only for Olly Lee to equalise for the hosts at St Andrew's and take the contest to a shootout, which Hughes's side eventually won 4-2.
The Welshman applauded their Championship opponents for their persistence, but was frustrated that his side surrendered their lead on more than one occasion.
"Credit to Birmingham, I thought they were magnificent," Hughes told reporters. "They kept on going and they were a credit to their manager and their club. With them being down to 10 men we got into a position where we are 3-1 to the good with 20 minutes to go, we should see the game out quite comfortably.
"But for whatever reason - maybe we lost a little bit of focus and ally that to the attitude and endeavour of Birmingham - we allowed them back into the game. I'm sure the neutral thought it was a fantastic game, from my point of view it was awful.
"You think the game's done and dusted and it should be, we need to be able to see out games in those circumstances quite comfortably. We showed an unbelievable amount of naivety in terms of what was required to put the match to bed."
Stoke are one-time League Cup winners, having lifted the trophy in 1972.