Dallas was reduced to 10 men just 10 minutes into the game as Je-Vaughn Watson was sent off for a high kick on New York's Tim Cahill.
From there, it was always going to be difficult for Dallas to take something from the game, although the hosts did have a penalty kick from Blas Perez saved in the second half before Wright-Phillips netted the winner.
New York has now earned 10 of a possible 12 points from its last four games, while Dallas lost for the third time in four matches.
The game took a dramatic turn in the opening 10 minutes when Watson received a straight red card for a kick to the groin of Cahill, leaving Dallas with 80 minutes to play a man down.
Cahill went close in the first half with a free kick that missed the left post by inches, but Dallas was handed a golden opportunity to take an unlikely lead.
Danny Garcia got down the left wing and into the penalty area where he was brought down by Cahill.
However, Perez stepped forward and had his effort saved by goalkeeper Luis Robles, paving the way for Wright-Phillips to put New York ahead.
Dax McCarty played Thierry Henry into the area on the right and he squared the ball to Wright-Phillips for a simple finish and three points for New York.