Now the Union (11-8-15, 48 points) must watch the scoreboard in hopes of avoiding a midweek wild-card game. Sporting Kansas City (48 points), the Columbus Crew (47), and the Houston Dynamo (46) are in the running for the top three spots, and all play their final game this weekend.
The Red Bulls finished 10-8-16 for 46 points and can't secure one of the top three spots.
"It is a good lesson this week - not letting them dictate the pace and rhythm of the game," Union defender Danny Califf said.
Trailing by 1-0 at halftime, the Union were much more aggressive in the second half.
In the 67th minute, Union defender Carlos Valdes was called offside on a Sebastien Le Toux free kick that went into the net, nullifying the goal.
"I didn't really see it," Le Toux said. "It was a tough break."
At halftime the Union inserted Danny Mwanga and Freddy Adu, and both gave the offense some life. They replaced 16-year-old Zach Pfeffer and Amobi Okugo.
In addition, Roger Torres, who entered in the 70th minute as the third and final Union substitute, gave the offense some life.
"We need to play the game the way we know," Union team manager Peter Nowak said. "The experience of the other team was a crucial factor in the whole thing."
New York struck first, in the eighth minute, on an own goal, although Dane Richards provided the damage. Richards headed a Jan Gunnar Solli cross off the far post. The ball deflected off the back of Mondragon and into the net.
It was an outstanding cross by Solli, one that went between Union defenders Valdes and Sheanon Williams.
The Red Bulls played without 14-goal scorer Thierry Henry, who was suspended after receiving a red card in Saturday's 2-0 loss to Kansas City. Still, the offense didn't lack chances in the first half.
In the 17th minute, Mondragon had to make a kick save on a breakaway by Richards that was shot just inside the box. Richards was sent on goal by Luke Rodgers.
The Red Bulls had another scoring chance in the 43d minute when Mondragon was again called to stop a breakaway, making a sliding save on Joel Lindpere.
Note. Beginning next season, 17 Union games will be broadcast live on WIP-AM (610). The station will partner with the team to produce a weekly, one-hour soccer show that will run through the end of the season in November.
Contact staff writer Marc Narducci at 856-779-3225, firstname.lastname@example.org, or @sjnard on Twitter.