Cooper's first goal came on an 11th-minute penalty kick following a takedown of Tim Cahill in the Philadelphia box.
Henry doubled the advantage in the 35th when he cleanly beat Philadelphia center back Carlos Valdes and rolled a shot past Union goalie Zac MacMath for this 15th tally of the year.
Cooper capped the scoring in the 57th minute for playoff-bound New York, which finished 2012 at 16-9-9 for 57 points and improved its chances for a Top 3 seed in the Eastern Conference.
The Union ended at 10-18-6 and will miss the post-season after making the playoffs in 2011.
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