Brady’s goal arrived just when it looked as if Ireland were heading out due to a combination of Italy goalkeeper Salvatore Sirigu and Romanian referee Ovidiu Hategan.
Ireland had to win the match at the Stade Pierre-Mauroy against an Italian side already assured of first place in Group E and got the win they needed to set up a last-16 tie against hosts France.
With top spot in the group sewn up and with one eye on their last-16 tie against Spain next Monday, Italy coach Antonio Conte made eight changes to his side.