New York had another decent opening midway through the first half when Henry curled a strike from outside the box just wide of the post.
The Red Bulls came close to equalising five minutes after the break through Wright-Phillips, who saw his header come back off the post.
They were soon back on level terms when Henry headed home from Eric Alexander's cross in the 53rd minute.
The Frenchman grabbed his second of the night fourteen minutes later with a strike into the bottom corner, before Wright-Phillips added a third goal in the 74th minute.
Andres Romero pulled a goal back for the visitors five minutes later, but, in the final minute of the match, Wright-Phillips scored a fourth for New York to seal the three points.