It was also a memorable moment for Brady, a 50-year-old Australian who replaced the sacked Darren Moore in January and has spent his managerial career in the shadows with Brackley and Northampton.
“I’m trying to let it sink in at the moment. It was tremendous from the boys, we had to suffer at times,” Brady said.
“I couldn’t believe we were in the last 16, I told my kids we were in the last 32 by mistake.
“It’s nice to create history, isn’t it!”
At Craven Cottage, Ross Stewart’s ice-cool penalty in stoppage time sealed Southampton’s 1-0 upset.
It was a bitter loss for Fulham manager Marco Silva, who paid the price for a selection gamble that backfired.
Silva made nine changes following a disappointing 1-0 defeat against West Ham in the Premier League earlier this week, but his switches failed to produce the desired result.
Southampton won it in the first minute of stoppage time when Finn Azaz was fouled by Fulham defender Joachim Andersen and Stewart drilled his spot-kick past Benjamin Lecomte.
Relegated from the Premier League last season, they are through to the FA Cup quarter-finals for the first time since 2022.
Southampton’s success served as a fitting tribute to the greatest day in the club’s history.
Their players wore a yellow commemorative kit to mark 50 years since they wore the same shirts when Bobby Stokes sealed second-tier Southampton’s shock 1976 FA Cup final win against Manchester United at Wembley.
Southampton’s current crop still have a way to go to emulate the achievement of Lawrie McMenemy’s men.
But Tonda Eckert’s side are unbeaten in 10 games in all competitions as they chase FA Cup glory and a place in the Championship play-offs.
“You see how much it means to the club,” Eckert said. “It’s been 50 years since we won it for the last time.
“When we arrive on the coach and you look and the whole street is full of Southampton supporters and behind the goal is full of Southampton supporters, you can definitely say we feel the importance of it.”
Later on Sunday, Leeds host second-tier Norwich, while West Ham meet Brentford tomorrow.
Arsenal, Chelsea, Manchester City and Liverpool booked their quarter-final places over the weekend.
