However, they were quickly pegged back as Dillon Serna found himself in possession on the right-hand side of the box before cutting back onto his left foot and tucking his effort past a helpless Zac MacMath.
Colorado tried to respond just after the break when Nick LaBrocca's shot was inches away from nestling in the top corner.
Philadelphia may have been forgiven for thinking that they had secured victory on 74 minutes when Andrew Wenger netted for the hosts, but Michael Lahoud conceded a penalty moments later for his dangerous challenge on Jared Watts, for which he received a red card.
With four minutes to go, the Union support were stunned when Deshorn Brown rifled his effort home to earn the Rapids an unlikely draw.
The result pulls the Union into sixth in the Eastern Conference, while Colorado are third in the Western standings behind Seattle Sounders.