Luke Rodgers scored the goal that let New York snap an eight-game winless streak and move into the final playoff position.
Dallas also missed wingers Brek Shea and Marvin Chavez to suspension.
New York was without three defensive mainstays, including Rafael Marquez, through a combination of injury and suspension. Dallas took advantage early on, with Maicon Santos guilty of spurning two early chances in the box.
Rodgers scored the only goal in the 36th minute. Juan Agudelo passed a low cross to the far post where the Englishman bundled the ball past Kevin Hartman.
"It doesn't seem like we're playing with the same type of intensity and we're not having as much success as we had perhaps in the middle portion of the season," Hartman said. "I don't know if it's because we have different personnel on the field, if it's because we're lethargic, we're tired. I don't know exactly what it is. I know we need to hurry up and rectify the situation as quickly as possible."
Joel Lindpere had the best chance of the second half when a neat cutback set him free in the box, but he skied his shot with Agudelo open in front of goal.
The win moves the Red Bulls into the last playoff position in the table.
"Hopefully now we can build on this, even if we had to defend a lot today" New York coach Hans Backe said. "We didn’t have that much of an attacking game, maybe two or three good chances. Overall it was a good game for us. Defend, defend, defend, and play a little bit dirty. We like to get these three points."