The Republic of Ireland lost the tie 2-1 on aggregate after Henry clearly handled the ball in the run up to the extra-time goal from William Gallas.
Television footage later showed a somewhat embarrassed Henry sitting on the pitch next to Dunne after the final whistle. He appeared to say the words, "I cheated."
Dunne later confirmed that was the case.
"He told us we deserved to win. How is that supposed to make me feel? It makes me feel worse," the Irishman told The Independent.
"He's admitted he cheated. We should have won. He just said, 'That's it'. He just said he handled it, he didn't mean it.
"Looking at it, it's quite obvious he did mean it. It's there for everyone to see and they're not going to change it now. So what can we do? They're going to the World Cup and we're not. That's it."
The Villa centre-back also expressed a certain amount of cynicism about how FIFA had organised the play-offs.
"FIFA will probably be happy. Yet again the big decisions have gone for the bigger team," he said, according to The Daily Telegraph.
"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."