Sporting (15-7-6) extended its unbeaten streak to six games, while Houston (12-7-10) has just one win in its past six matches.
The Dynamo went up 1-0 when Brad Davis scored in the 58th minute on an assist from Boniek Garcia, and appeared headed for the win until Sapong headed home Graham Zusi's cross. It was the 13th assist of the season for Zusi, tying him for the league lead with Seattle's Mauro Rosales.
Sapong scored on his career-high ninth shot of the night. Two of his attempts were saved by Houston's Tally Hall in a scoreless first half, but Sapong's goal denied Hall what would have been his MLS-best 12th shutout.
Houston was without coach Dominic Kinnear and midfielder Calen Carr, who both were ejected from last week's 1-0 win over Real Salt Lake. Carr had two goals when the Dynamo beat Sporting 2-1 in Houston on July 18.