The rare weekday afternoon game was played in record-breaking 106-degree temperatures, the hottest July 18 ever recorded in New Jersey.
In the 71st minute, Henry blasted a left-footed shot that went off the far right goalpost and into the net. It was Henry's 10th goal of the season.
Chicago (9-7-4) had owned a 6-1-3 record over the Red Bulls since 2008.
Bill Gaudette, playing in his second game with the Red Bulls (10-5-5), registered the shutout but did not have to make a save.
Newcomer Sebastian Le Toux made a perfect 35-yard cross to Henry. The French star settled the ball with his chest, then put it to his left foot. The shot was from a tough angle, but Henry managed to get it high off the post and past Sean Johnson for the game's lone goal.
The Red Bulls survived a scare in the 84th minute, when Gonzalo Segares' header appeared to go past Gaudette, but an alert Brandon Barklage headed it out of danger to preserve the lead.
In the heat and humidity, the first half was played at a snail's pace, with many of the players standing around instead of making any long runs with the ball.