A defensive blunder by the Fire gave the Red Bulls a 1-0 lead in the 16th minute.
Sean Johnson hesitated on a ball played into the box by Mike Grella and Matt Polster slid and tapped into his own net from the edge of the six-yard box.
Chicago leveled matters in the 35th minute when Accam picked up a second ball after Michael de Leeuw won a 50-50 challenge with Ronald Zubar and ran into space. He blasted a low shot to the left of Red Bulls goalkeeper Luis Robles from 24 yards out for his fifth goal of the season.
Polster made amends for his earlier gaffe by latching onto a whipped in ball by John Goossens and putting his diving header past Robles from the edge of the six-yard box in the second minute of first-half stoppage time.
However, Bradley Wright-Phillips leveled matters in the 90th minute when he tapped in a cross by Connor Lade to help the visitors escape with a 2-2 draw.