The French, who celebrated their first knockout-stage win in the competition since winning Euro 2000, were stunned when Paul Pogba gave away a clumsy spot-kick and Brady converted it to score the tournament's fastest goal at Parc OL.
A disjointed France came to life in the second half as the introduction of winger Kingsley Coman injected pace and bite up front, with Griezmann heading home the equaliser in the 58th minute after a superb cross by Bacary Sagna.
Wales beat Northern Ireland 1-0 in Euro 2016 last 16
The 25-year-old turned the match on its head three minutes later after Olivier Giroud headed a long ball into his stride for a clinical finish. Ireland had centre back Shane Duffy sent off midway through the second half after he hauled down Griezmann who was through on goal.
France will play either England or Iceland for a semi-final place.
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