Former Germany star Schweinsteiger headed home David Accam's perfectly flighted cross from close range after 17 minutes to put Chicago ahead.
But the veteran midfielder was denied a fairytale start to his MLS career when Matteo Mancosu equalised for Montreal in the 61st minute.
A stormy encounter then saw Chicago midfielder Juninho sent off in the 71st minute after picking up a second yellow card for unsporting behavior.
Both sides were reduced to 10 men however in the 80th minute when Montreal's Victor Cabrera was given a straight red for bringing down Argentine forward Luis Solignac as he went through on goal.
Montreal looked to have grabbed a late winner when Ballou Jean-Yves Tabla lashed in a shot from outside the area to make it 2-1 in the 90th minute.
But Chicago grabbed a share of the points in the third minute of stoppage time when Solignac controlled with his chest on the edge of the area before sweeping home a low finish to make it 2-2.
The draw leaves Chicago in seventh place in the Eastern Conference standings, with five points from four games.