Chicago's equalizer came when Sebastian Grazzini put a ball into the penalty area and Oduro was able to redirect it past goalkeeper Donovan Ricketts on a lunging attempt.
The Fire (0-0-1) thus secured a point in their 2012 regular-season opener.
The game -- Montreal's first home contest as a member of the MLS -- was played in front of 58,912 at Olympic Stadium, a new attendance record for professional soccer in Canada.
The Impact generated a pair of chances in the first half before grabbing a brief lead in the 56th minute on a goal by captain Davy Arnaud.
Notable deaths of 2014 [PHOTOS]