Union, which is fifth in the Eastern Conference, went behind in the 10th minute when Przemyslaw Frankowski headed in Boris Sekulic's cross.
Chicago was reduced to 10 men when Wyatt Osmberg received a straight red card in the 34th minute, after VAR review, and Philadelphia capitalised from the ensuing free-kick.
German defender Kai Wagner scored the equalizer with a shot which went under the jumping Chicago wall.
Daniel Gazdag spurned a chance for Philadelphia with a header which went wide in the 52nd minute, while Union substitute Ilsinho fluffed a late volley.
Philadelphia has now won only one of its past seven games.
Nashville edged its way up to third in the Eastern Conference with a 1-1 draw at lowly Toronto.
Omar Gonzalez had headed Toronto ahead from a 19th-minute quick free-kick, but Robert Castellanos levelled before half-time from a long throw-in.