Cristian Casseres Jr. gave New York a seventh-minute lead before Luke Berry equalized in the 18th minute. The match appeared to be headed for a draw until Sean Nealis won it in the 87th minute.
The victory sends the Red Bulls into sixth place in the Eastern Conference standings while the Crew remain five points out of the playoffs with three matches remaining.
Story of the match
For the second week in a row it was Cásseres scoring on an early set piece. Tom Edwards sent in a corner that the Venezuelan was completely unmarked on and he cushioned past Columbus goalkeeper Eloy Room to give the Red Bulls a 1-0 lead.
Columbus equalized as Derrick Etienne found Lucas Zelarayan in space and the Argentine swung the ball to Moreira, who crossed for Berry and he finished past a helpless Carlos Coronel in the 19th minute.
Columbus would control much of the remainder of the half and Moreira's long cross found Etienne, forcing Coronel into a brilliant diving save just before halftime.
On the resulting corner kick, Berry attempted an overhead kick on the loose ball, but the shot was blocked.
In the second half, the Red Bulls were awarded a penalty when Wiki Carmona was chopped down by Marlon Hairston on a lunge into the box. Patryk Klimala stepped up to the spot, but pulled his shot wide left of Room’s goal.
The Red Bulls nearly took the lead in the 79th minute, but Room parried Kyle Duncan's shot wide. After a turnover, Carmona got free, but his shot from 22 yards blazed over the bar.
Nervy moments for New York came shortly after when a misplayed Coronel clearance forced the goalkeeper to make a slide tackle near the midfield line.
The visitors snatched a winner late on. After retaining the ball in the 88th minute following a possession-induced corner kick, New York cycled the ball around to Cásseres, whose cross was headed by Klimala back across goal into the path of Nealis, who finished past Room.
Man of the match: Sean Nealis
Named to the MLS team of the week, his leadership, drive and passion were on display in this one. A big goal and a big defensive presence helped his side above the playoff line.