Kleberson's goal came just about at the end of six minutes of stoppage time and extended Philadelphia's winning streak to two.
The Union (12-10-9) moved past Chicago into fifth place in the Eastern Conference.
Philadelphia seemed headed for a third draw against Toronto FC this season, especially when Fabinho was issued a red card in the 77th minute, sending it a man down.
Kleberson, who had been acquired in a trade for Freddy Adu, was a late substitution and scored his first MLS goal.
Toronto FC (5-16-11) has the second-fewest wins in MLS, behind only D.C. United (3-22-6).
Philadelphia has three games remaining and is searching for its second playoff berth in the four-year history of the franchise.