Philadelphia and Toronto appeared headed for a draw until Kleberson pulled off a bit of magic to take the win.
The Union had a number of chances in either half, with Jack McInerney and later Antoine Hoppenot coming within inches of finding the net.
Philadelphia went down a man when a red card was shown to Union wide man Fabinho in the 77th minute for a stomp into the back of Toronto's Alvaro Rey.
Despite the disadvantage, the Union came up with a huge win. Kleberson scored a vital goal deep into stoppage time, blasting a free kick past a stationary Joe Bendik.
Philadelphia (12-10-9) moves into the fifth and final playoff spot with the victory. TFC (5-16-11) remains eighth in the East.