Instead, the reigning U.S. Open Cup champions were level with the defending MLS Cup title-holders, ending with a 1-1 draw and Houston's elimination from the big-time international tournament.
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The Dynamo had the disadvantage of a 1-0 deficit in the series after last week's first leg in Houston, where an own goal gave Columbus the aggregate lead.
On Tuesday, Houston conceded a 40th-minute goal by the Crew's Cucho Hernandez, who got a right-footed shot past Dynamo keeper Steve Clark.
With time running out, the Dynamo scored in the 90th minute when Griffin Dorsey converted a penalty awarded to Houston after a handball was called against Columbus.
Five minutes of stoppage time was added, but the Dynamo never got another chance at scoring after the club drew two bad-foul yellow cards.
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The Crew held the 2-1 aggregate advantage after both legs, enough to advance to the quarterfinals and oust the Dynamo.
Houston's focus now shifts to the Major League Soccer regular season. The Dynamo are winless after two matches, owning a draw, a loss, and just one point for the season.
The Dynamo's next MLS match is on Saturday when they host the Portland Timbers at Shell Energy Stadium.