The big Brazilian has had a difficult start to life at Elland Road, losing his Premier League starting place to Karl Darlow but he was the main reason the Whites came through a severe fourth round test against their Championship opponents.
After more than two hours of thrilling entertainment, Perri denied Tommy Doyle in the spot-kick tie-breaker before Patrick Roberts blazed over enabling Sean Longstaff to rattle in the decisive kick as Leeds won 4-2 on penalties after the sides were locked at 1-1.
The result was hard of the Blues who were the better side but were punished for not taking their chances as Leeds struggled to match the home side’s physicality.
Both managers made six changes, but it was Birmingham who soon had Leeds on the back-foot in a storming start.
Perri brilliantly thrust out a huge paw to touch Jay Stansfield’s fiercely struck half-volley on to the woodwork to keep Leeds level.
The visitors could not get any meaningful possession as they were out-muscled in the middle of the park, United’s only meaningful effort being a 25-yard shot by Longstaff which was saved by Ryan Allsop.
Leeds were relieved – and Birmingham disappointed - to reach the break goal-less.
The Yorkshire club’s manager Daniel Farke brought on Ethan Ampadu for full debutant Facundo Buonanotte, who had endured a nightmare half, and Leeds immediately looked more settled.
Four minutes after the resumption, Noah Okafor’s slick turn opened up the Birmingham defence, played in Lukas Nmecha who blasted in from the edge of the box.
Birmingham refused to cave in and came roaring back to put United under pressure. The home team’s crosses into the box being far more effective than Leeds’ scruffy efforts.
United have struggled to keep clean sheets and have let in late goals all season so it was no surprise when Birmingham deservedly drew level on 89 minutes.
A corner was cleared to Roberts whose left-foot missile of a shot was deflected off James Justin into the corner of the net.
Both sides pushed for a winner in extra time, the Blues coming nearest when Ibrahim Osman’s effort came back off an upright to Kanya Fujimoto whose goal-bound strike was blocked by Jaka Bijol.
Leeds went on to hold their nerve in the penalty shootout. They were knocked out of the Carabao Cup on penalties at Sheffield Wednesday at the start of the season when they missed all their kicks but made no mistake at St Andrew’s.
Joel Piroe, Dominic Calvert-Lewin, Brenden Aaronson, and finally, Longstaff all netted as Leeds scraped through.
Fan favourites Ao Tanaka and Willie Gnonto were handed starts but they, alongside Buonanotte, failed to make a case for a place in the United’s Premier League starting line-up.
Farke said: “It was far away from champagne football … but we found a way to get through.”