Robin van Persie had put the Gunners on course to win the preseason trophy they created for a third successive year with a first-half headed goal.
But Arsenal, which lost a lead Saturday to draw with Boca Juniors, crumbled again. In the last five minutes, Henry passed to Roy Miller, whose cross was bundled into his own goal by Arsenal's Kyle Bartley.
The Red Bulls won the event because they topped the standings after having beaten Paris Saint-Germain 1-0 on Saturday.
As the Red Bulls players prepared to lift the trophy, Arsenal's players were booed off the pitch.