Martin O'Neill's side knew only a victory would secure a place in the knockout stage in France but had failed to find a way through the Azzurri's defence until Brady headed home Wes Hoolahan's cross in the 84th minute.
That sparked mass celebrations from the Republic players, bench and supporters under the closed roof at Stade Pierre Mauroy, which had largely been taken over by the Boys in Green.
Defeat made little difference to Italy, who already knew they would top Group C and made eight changes ahead of Monday's last-16 match with holders Spain.
But the Republic join them - and will face the hosts on Saturday - after one of the most famous wins in their history that started a huge party in Lille.
Competition: European Championship
Date: 22 June 2016
Stadium: Stade Pierre-Mauroy (Villeneuve d’Ascq)
Referee: O. Hațegan
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