Ireland knew that they had to beat the Azzurri to qualify from Group E and they did just that thanks to Brady, who climbed highest to divert a pinpoint cross from Wes Hoolahan - who had missed a glorious chance just moments earlier - beyond Salvatore Sirigu with just five minutes remaining.
It looked as if a much-changed Italy, who were already sure of topping the pool, were going to frustrate Martin O'Neill's side, but Brady's intervention means they will now meet Les Bleus in Lyon Sunday.
Speaking at a news conference after being named man of the match, Brady said: "I lost myself a little but couldn't help it, it's what dreams are made of.
"It's a proud night for myself and fantastic for the team. People talk of an out-of-body experience with these goals and I was lucky to experience it. I'm stuck for words, still.
"The chance Wes Hoolahan had was a massive moment but it shows the character and playmaker he is to pick the ball up two minutes later and put it where I ran, right on my head. I'd have done well to miss.
"It was our last chance, I said we had players to hurt them and it shows the team spirit. We had it all, we rode our luck at times but stuck at it and put in a great performance. Roll on France."
Brady added: "We've shown some fantastic results to get here. It was hard against a relentless Italian side but I don't think we've got anything to fear, we know they're a top-class team but if we click we can hurt anybody. We can go as far as we allow ourselves in this tournament.
"We watch a lot of their players on the world stage, we know a bit about them. We have to prepare as much as we can but we'll concentrate on us and do our homework. But we know they're not one to be taken lightly."