Houston (5-10-3) ended a club-record five game losing streak.
Davis was awarded the penalty kick after New York's Chris Duvall was whistled for a handball.
Houston's Giles Barnes opened the scoring in the first minute, heading home Kofi Sarkodie's cross to end the Dynamo's goal drought at 479 minutes.
New York (4-5-8) tied it in the 13th minute when Bradley Wright-Phillips ran down a long ball from Eric Alexander and put home a one-touch effort. Wright-Phillips made it 2-1 in the 72nd with his MLS-leading 14th goal of the season, redirecting a cross from Ambroise Oyongo Bitolo.