Leading 1-0 through Lukas Nmecha, Leeds were taken to extra-time after Patrick Roberts equalised just before the 90 minutes.
With the scores locked at 1-1 at the end of extra-time, the tie went to penalties, which saw Perri decisive as he denied Tommy Doyle before Roberts blazed his kick over the crossbar in the shootout.
Afterwards, the Argentine said: "It was an incredible fight from the team.
"We had a very difficult game, it's always difficult to play against a team like Birmingham, so congratulations to the whole team for the fight and we got to the next round."
On preparing for penalties, Perri admitted: "We had a video but unfortunately for the first two penalties because they didn't have the number on the shirt it was difficult for me to see.
"And the third one we didn't know where he shot so I went on instinct and happy to help the club."