Chivas goalie Dan Kennedy had stopped eight attempts in regulation time and another Houston shot hit the post before Hainault scored the game's only goal.
Chivas' Juan Pablo Angel hit the post with a shot late in the first half.
The victory is the first in a season opener for the Dynamo since 2006.
Houston lost the MLS Cup final last season 1-0 to the Los Angeles Galaxy.
2014: The Year in Music [PHOTOS]