McInerney's tally came on a header in the edge of the box off a pinpoint low cross from Sebastien Le Toux.
The early score was enough for the Union to improve to 5-4-3 on the season.
The Fire, meanwhile, couldn't dent Philadelphia goalkeeper Zac MacMath despite getting six shots on net.
They fell to 2-7-1 with the loss.
Celebrity Families of 2014 [PHOTOS]