The home side took the lead in the 10th minute when Brad Davis scored from the penalty spot after Giles Barnes was brought down in the box following a surging run into the area.
Houston almost doubled their lead when Barnes picked out Ricardo Clark in the six-yard box, but his headed effort was well saved by Matt Lampson in the Crew goal.
However, Lampson could do nothing to prevent Houston from doubling their lead in the 31st minute when Oscar Garcia robbed Agustin Viana and drove upfield before feeding Will Bruin, who slotted the ball into the net.
After the break Columbus got back into the game in fortuitous circumstances when Kofi Sarkodie turned the ball into his own net while attempting to tackle Ryan Finley.
Houston responded in spectacular fashion as they restored their two-goal lead through Cam Weaver, his 20-yard volley finding the top corner to seal the win for his side.