Trailing 1-0 from the first leg, the Irish levelled the tie when Robbie Keane found the net just before half-time in the Stade de France and there were no further goals during normal time.
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Les Bleus' progress was sealed right on half-time of extra-time when William Gallas turned the ball home from close range, although Henry had clearly handled in setting up his former Arsenal team-mate.
Aston Villa defender Dunne, who spoke to Henry on the final whistle, said: "He admitted afterwards he handled it, but it doesn't make me feel any better because we aren't going to the World Cup finals.
"FIFA will probably be happy. Yet again the big decisions have gone for the bigger team.
"There has to be a case for matches of this importance for a fourth official to look at replays and get the calls right. We were cheated tonight."