The Red Bulls (11-5-5) remain undefeated at home this season, improving to 7-0-3. New York is the lone MLS team without a loss at home.
Cooper, who did not start Wednesday's game against Chicago, scored off a pass from rookie Connor Lade in the 43rd minute, then added another header off a fine feed from Thierry Henry in the 58th minute. Cooper has 13 goals this season, four behind San Jose's Chris Wondolowski for the MLS lead.
Lade had two assists.
It was the second straight shutout victory for Bill Gaudette, who made two saves in place of injured starter Ryan Meara. Gaudette has started three straight games with two wins and a tie.