New York (7-3-1) found itself down 2-1 in the second half, but rallied after the Union's Freddy Adu was sent off with a second yellow for diving, leaving Philadelphia a man short.
Markus Holgersson headed in the tying goal in the 68th and 10 minutes later Cooper converted into an empty net when Union goalie Zac MacMath mishandled a ball.
Joel Linpere also scored for New York, which climbed to the top of the Eastern Conference -- one point ahead of Sporting Kansas City -- despite playing without leading scorer Thierry Henry.
Philadelphia (2-6-1) got a pair of goals from Lionard Pajoy.
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