Frankowski's goal was enough to earn Chicago (3-9-4, 13 points) its first point on the road in six attempts despite playing the last two-thirds of the game down a man after Wyatt Omsberg's dismissal.
Kai Wagner scored on a first-half free kick to level the game for Philadelphia (6-4-7, 25 points).
But the Union couldn't capitalize on their monopoly on chances and possession after the break, missing a chance to jump above New York City FC and potentially Nashville in the standings. They've now won only once in their last seven matches while slipping to fifth in the Eastern Conference.
Chicago started brightly and took the lead when Frankowski headed in Boris Sekulic's cross to the back post after 10 minutes. And the Fire continued to look promising in the match's early stages.
That changed in the 34th minute when, following a video review, referee Michael Radchuk ruled Omsberg had pulled down Cory Burke to deny an obvious goal-scoring opportunity.
Wagner drove the ensuing free kick under the leaping defensive wall and inside the right post, leaving goalkeeper Bobby Shuttleworth little chance.
Philadelphia had several second-half chances but struggled to put them on target. Burke dragged a shot and then flashed a header both wide of the left post in the opening five minutes of the half.
Jose Martinez finally tested Shuttleworth in the 80th minute, but the goalkeeper dove right to push away the well-struck effort from distance.
Alejandro Bedoya also had a one-time effort on a cross denied by Shuttleworth, and Kai Wagner fired an effort from distance over the crossbar.
Between several stoppages for Fire injuries, Radchuk added on more than 10 minutes of second-half stoppage time, but the Union appeared to run out of energy and ideas.