Gauci made a series of brilliant saves to help the League One side protect their 1-0 lead on Sunday, with Vale's Australian manager Brady admitting the win that's thrust them, wholly unexpectedly, into the quarter-finals is actually a distraction as they're bottom of the table and in a relegation scrap.
"It probably hasn't sunk in really. I'm in a little bit of shock about that result today," Brady smiled.
"It's a bit of a pain in the bum, to be honest! It is really, because what it's doing for us is just adding to our fixture list.
"We lost one of our best centre-halves (Cam Humphreys) to injury and we haven't got the size of the squad to cope with the amount of games at the moment.
"It's a privilege but it's tough as well."
But the Newcastle-born Brady, who only came to the club a couple of months ago after a previous spell managing Northampton, is guaranteed folk-hero status at Vale Park after orchestrating the victory over Sunderland, who are 57 places above Port Vale in the English soccer pyramid.
They're now in the last-eight for the first time in 72 years alongside giants like Liverpool, Arsenal, Manchester City and Chelsea, courtesy of New Zealander Ben Waine's looping first-half header.
It was a great day for Waine, who was born and raised in Wellington but who was a Newcastle fan as kid because his family hail from England's north-east. Newcastle, of course, just happen to be Sunderland's deadly enemies.
Which team would 51-year-old Brady fancy facing in the next round? "I wouldn't have a clue because I'm only thinking about Bradford on Wednesday," he smiled.
"Actually, I couldn't believe we were in the last-16 - I told my kids we were in the last 32!"
"It's nice to create history, isn't it?"
His men had only secured their place in the fifth round by also shocking Championship side Bristol City five days earlier -- with Waine scoring the winner in that match as well.
But this time, he wasn't the only Oceanian hero on the pitch as Gauci, the 25-year-old who's on a season-long loan from Aston Villa, made saves which Brady called "incredible".
It wasn't the only shock of the day, as Ross Stewart's 91st-minute penalty sealed a 1-0 victory for Championship side Southampton at Premier League Fulham.
Leeds broke the sequence of upsets by beating Norwich 3-0 at Elland Road.
