The Crew (10-7-15, 45 points) tied Inter Miami for the final playoff spot in the Eastern Conference with two matches left. Miami holds the tiebreaker.
Etienne scored the winner against his former team from close range to deny the Red Bulls (14-11-8, 50 points) assurance of a home playoff game.
Frankie Amaya's goal in the 53rd minute provided New York with the lead.
Amaya's goal was the result of a giveaway. He fired from 28 yards and the ball deflected off a Crew player for his first goal of the season.
The Crew's first shot on goal -- by Etienne -- was saved by Carlos Coronel in the 85th minute.
Also, Lucas Zelarayan (nine goals, 11 assists) did not start for the Crew and entered a minute after the Red Bulls' goal.
Columbus knew it needed points after Inter Miami defeated Toronto FC on Friday to gain the three-point advantage.
The Crew play at Charlotte FC on Wednesday to complete the game of July 30 that was suspended due to inclement weather. Columbus finishes at Orlando City on Sunday, the same day New York hosts Charlotte.
Columbus didn't offer much of an offense in the first half despite their tenuous playoff hopes.
They managed just one off-target shot through 45 minutes -- despite 69 percent of possession -- while New York was aggressive.
Daniel Edelman rang a shot off the post in the 17th minute, the first of five shots for the Red Bulls in the opening 45 minutes. Amaya just missed a long attempt in the 41st and New York's first shot on goal came in the 44th minute by Andres Reyes.