Philadelphia (0-3-2) failed to convert a one-on-one opportunity with the goalkeeper in the 88th minute for a two-goal cushion and Sporting KC (2-1-2) took advantage with goals in the 91st and 94th minutes.
Anibaba headed in the equalizer and Nemeth redirected a corner kick for the go-ahead goal.
It's the seventh time in MLS history that a team has trailed entering stoppage time and prevailed.
Philadelphia got on the board in the 3rd minute when a corner kick sailed over goalkeeper Luis Marin and bounced off Jacob Peterson for an own goal. It was the earliest goal ever conceded by Sporting KC.
Sporting's Dominic Dwyer headed in Benny Feilhaber's corner kick in the 16th minute for his first goal of the season. Philadelphia retook the lead three minutes later when Fernando Aristeguieta headed home Sebastien Le Toux's free kick.