The Red Bulls opened the scoring in the seventh minute when Richards took it down the right side of the box and tapped the ball to a sliding Macoumba Kandji, who directed it to the right of goalkeeper Joe Hart.
"Dane can run, and when I saw him I took off and found the blind spot in the defense," Kandji said.
The Red Bulls put pressure on Manchester throughout the first half, firing off seven shots to Manchester's two.
Thierry Henry, who scored his first Red Bulls goal in Thursday's loss to Tottenham, took three shots. Two were blocked by defenders and one sailed wide right. The French international star came out of the game for the second half.
The Red Bulls's next MLS game is Saturday in Houston.
"The most important thing is the Houston game," Henry said. "I'm trying not to think too much about it."
Man City's Jo tied the game in the 55th minute, tucking a quick shot into the upper right corner off a feed from the wing.
Neither goalkeeper Bouna Coundoul nor his defense had a chance to react.
Then, in the 68th minute, Kandji returned Richards' favor by setting him up with a perfect backpass that allowed Richards to keep his momentum and blast the ball off the crossbar and in.