Anangono ran down the right endline and headed Mike Magee's 42-yard free kick past Philadelphia goalkeeper Zac MacMath.
The Fire are winless in five matches (0-1-4), while Philadelphia (1-1-3) collected its second tie in a row.
Chicago was awarded a penalty kick just before the 5 minutes of stoppage time elapsed, but MacMath stopped Magee, the 2013 league MVP, on the original kick and a close-in rebound to preserve the deadlock.
Philadelphia scored twice in a seven-minute span to erase Chicago's early lead.
Midfielder Maurice Edu connected from 6 yards out in the 32nd minute after Bakary Soumare froze, believing the Union had been called for a hand ball after Vincent Nogueira's shot was blocked by the Fire's Jhon Hurtado.
Edu stepped up and beat goaltender Sean Johnson from the edge of the 6-yard box for his first goal in MLS play. The 27-year-old Californian had played in Europe the last four seasons.
The Union took the lead on Leo Fernandes' 30-yard free kick in the 39th minute. His shot from the right side of the field cleared a crowd of players and bounced in front of Johnson, who misplayed the shot, handing Fernandes his second goal of the season.
The Fire opened the scoring on Magee's first goal of the season in the 16th minute. Quincy Amarikwa stole the ball from Philadelphia defender Amobi Okugo, ran to the left corner and fed Magee, who was open 4 yards out.
