The former Barcelona and Arsenal star passed to Angel for goals in the 10th and 58th minutes, twice putting New York ahead in his first competitive match since France's first-round World Cup exit in June.
However, Brian Mullan scored in the 90th minute to lift the Dynamo (5-8-5) to their fourth tie in five matches. Brad Davis also scored for Houston, which is winless since May 22.
New York (8-6-3) got its third tie in four matches to remain in second place in the Eastern Conference.
Angel put New York ahead 1-0 after getting a pass from Henry and hitting it into the upper left net. Davis tied it in the 24th minute on a penalty kick, hitting it into the right net. Houston was awarded the kick after Joseph Ngwenya was tripped from behind in the box by Carlos Mendes.
Davis missed another penalty kick in the 39th minute after Brian Ching was tripped in the upper box by Tim Ream.
Angel scored in the 58th minute to put the Red Bulls ahead again after getting a pass from Henry, dribbling to the right box and firing in the upper left net from 10 yards out.
Mullan tied the score on a header from the left box into the left side of the net after getting a pass from the right from Andrew Hainault.
Houston played a man down for the entire second half after Lovel Palmer was sent off with a straight red card for kicking Mendes in the shin when the two collided in the 44th minute.
Henry had several great opportunities to get his first MLS goal, having a shot saved by Houston goalkeeper Tally Hall, then the rebound into the right net kicked out by Ngwenya in the 31st minute and missing just wide of the far post on a breakaway in the 42nd minute.
Two fans rushed the field after Mullan's goal and jumped on Henry's back as he waited to kick off at midfield. After a long chase from one end of Robertson Stadium to the other, the fans were eventually captured by security.