Estonian midfielder Joel Lindpere had put New York ahead in the 61st minute with a close-range finish after smart work from Moroccan substitute Mehdi Ballouchy.
Dallas, last year's beaten MLS Cup finalists, rarely threatened Red Bulls' German goalkeeper Frank Rost in what was a disappointing performance with their highly-rated winger Brek Shea largely ineffective.
New York were reduced to 10 men in the 79th minute when Norwegian Jan Gunnar Solli was sent off for a tackle from behind on Daniel Hernandez.
But Red Bulls kept it tight and in the ninth minute of injury-time, Henry broke away and although his shot was parried out by Dallas keeper Kevin Hartman, the former Arsenal and Barcelona striker tapped the loose ball home.
The Red Bulls will now face the Galaxy, who finished the regular season with the best record, on Sunday at Red Bull Arena.
The second wild-card game features Colorado against Columbus on Thursday, while the conference play-offs start on Saturday with Real Salt Lake hosting the Seattle Sounders.
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