Chivas enjoyed large amounts of possession in the early stages and Nick LaBrocca tested Ryan Meara with a long-range effort in the second minute.
As the first half progressed the Red Bulls started to dominate the match and Henry could have made up for his earlier miss but the Frenchman was once again wasteful as he dragged his effort wide.
The half-time break seemed to invigorate Chivas and the visitors were ahead just two minutes after the restart when Juan Pablo Angel found space in the box to head in from a corner.
Chivas failed to build on their lead and New York levelled the match less than 10 minutes later.
Dane Richards played in Cooper who placed his shot powerfully into the bottom corner.
As the game entered its final stages, Henry once again found himself free in the box but the striker's disappointing day was epitomised by his failure to find the target with a header from close range.