Having to win Wednesday's match to progress from Group E, Ireland looked to be heading out as they went into the final five minutes at Lille's Stade Pierre-Mauroy with the game still goalless.
But then Brady appeared, running onto a Wes Hoolahan cross to score and secure a 1-0 victory that took Martin O'Neill's men through as one of the best third-placed sides.
The win set Ireland up for a rematch with the French, seven years on from the infamous Thierry Henry handball that knocked the Irish out in a World Cup qualifying play-off.
And Brady said the triumph, that followed a 1-0 win over world champions Germany in qualifying, should now stand Ireland in good stead.
"We have managed to beat some of the best teams in the world. We know (France) are a top-class team and they have got some unbelievable players but if we can stick together and click we can hurt anybody," said the Norwich City player, who would have merited his man of the match award even without his goal.
"I don't think we have anything to fear and I think we can go as far as we allow ourselves in this tournament."
For Ireland it was a first win in a competitive match against Italy since the 1994 World Cup and Brady's heroics mean he – at the age of just 24 – has secured himself legendary status among Irish fans.
"I can't really explain the feeling. You hear people who have scored these types of goals saying that it is like an out of body experience and I was lucky enough to experience it tonight," acknowledged the Dubliner.
Ireland's win in front of a sea of green in the Lille stands came after they were held to a 1-1 draw by Sweden in their opening game and then suffered a comprehensive 3-0 defeat to Belgium.
It also justified manager O'Neill's decision to make four changes to his starting line-up, including completely changing his central defence, where Shane Duffy of Blackburn Rovers held his nerve brilliantly in his first competitive international.
"You would've thought that Duffy was playing in his 35th international. He was terrific and so was the whole team," said O'Neill.
"There wasn't a player out there in an Irish shirt who didn't perform heroically."
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