An Austin own goal forced by John Tolkin and Dante Vanzeir gave the Red Bulls an early lead, but that was cancelled out by a Sebastián Driussi goal in the 44th minute.
It was Austin's only shot on target all match, but it was enough to earn the road team the draw.
The Red Bulls are now winless in seven of their last eight league matches. With only five league matches to go, the Red Bulls currently sit five points below the playoff line in the Eastern Conference.
New York will now face a quick turnaround and a trip to Atlantic Cup rivals D.C. United, who currently hold the last playoff spot in the Eastern Conference, on Saturday, September 23 at 7:30 pm ET.