They left the field to restless boos after extending their winless streak to six games. The Union (1-2-5) have gone more than a month without a victory.
"A tie is the most frustrating result," defender Sheanon Williams said. "Maybe not today, but we're mostly outplaying teams and not getting the win."
After the way they opened the game, a draw was perhaps fortunate. The Dynamo (2-3-1) outshot the Union, 12-1, in the first half and hit a post.
The Union, whose starting lineup did not include Sebastien Le Toux or Leo Fernandes, did not take their first shot until the 37th minute. The starters included Andrew Wenger and Conor Casey attacking together for the first time.
"We all looked ourselves in the mirror and we realized [the first half] wasn't good enough, especially at home," Union defender Amobi Okugo said. "We didn't even have a shot on goal. That's unbelievable to be at home and not have a shot."
The Union mostly controlled the second half.
Casey's plant leg slipped on a chance he should have buried. In stoppage time, Wenger lofted a one-time chance from directly in front over the crossbar.
Houston, which had come in going more than 270 minutes without a shot on target, had the lion's share of the chances.
The Dynamo had 12 shots in the first half, including one by Will Bruin that rang off the post to the right of Union goalkeeper Zac MacMath.
"We should have pressed them from the start. We're not going to give an opponent time and space in our own building. That's not who we are," Union manager John Hackworth said. "As frustrated as I was at halftime, after the second half I think there are . . . positives."