Henry's goal, coming after Marvin Chavez's second goal put Dallas up 2-1 in the 79th minute, came off a direct kick from newly-acquired Dax McCarty. Henry blasted it in from just to the left of the box, beating goalkeeper Kevin Hartman to the wide side.
The tie put the Red Bulls (6-5-12) a point behind Eastern Conference leader Philadelphia, while Dallas (11-5-6) remained three points behind West-leading Los Angeles.
U.S. national team member Juan Agudelo's fifth goal of the season in the 39th minute put the Red Bulls up 1-0. Dane Richards ran the ball down the right side and fed it into the forward, who chipped the ball in a high arc over goaltender Kevin Hartman and into the far left side.
Chavez scored in the 51st minute to tie the game.
The Red Bulls are now 20-0-9 when scoring first in the two years under head coach Hans Backe. New York is also 2-4-10 in its last 16 games.
The Red Bulls played the final 13 minutes with 10 men thanks to a second yellow card issued to Richards in the 80th minute. He will miss the Red Bulls' next match, at Real Salt Lake on Aug. 6.