McInerney headed home a corner kick from Sebastien Le Toux for Philadelphia (5-4-3), which beat the Fire 1-0 for the second time in a week. The Union also won 1-0 at Chicago last Saturday.
The Fire played short-handed for much of the second half after Wells Thompson received his second yellow card - and an automatic ejection - in the 62nd.
Chicago (2-7-1) lost its third game in a row to fall to 0-4-1 on the road.
The Union improved to 2-2-2 at home with their first victory in Philadelphia since a 1-0 win over New England on March 16.