The Dynamo (3-1-3, 12 points) had to play the final 15-plus minutes of the match with just 10 men after center back Teenage Hadebe was assessed a red card for yellow card accumulation.
The draw snapped a two-game winning streak for Houston, which had no answer for Portland goalkeeper Aljaz Ivacic.
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Steve Clark, signed by Houston in the offseason after he led the Timbers (2-2-4, 10 points) to a berth to the MLS Cup last year, matched Ivacic to keep the game scoreless. Clark had two saves in racking up his second shutout and the 46th of his MLS career.
Ivacic's shutout, which featured three spectacular saves among his four stops, was the second of the campaign and second of his MLS career.
Houston produced the two best scoring chances in the first half, but Ivacic was up to the task on both. First, Darwin Quintero had a one-on-one opportunity and then a rebound in the seventh minute but his attempts toward the bottom-left corner were turned away.
Then, in the 19th minute, Fafa Picault was sent on a breakaway by a pass from Quintero but was stopped when Ivacic came well off his line to get a right foot on it and deflect Picault's shot.
Portland's only shot on goal in the first half came in the 39th minute, when Cristhian Paredes' right-footed shot from outside the box was saved by Clark.
Ivacic made the save of the night in the 64th minute, diving to his left at the last moment to get a hand on Picault's header after a pass into the box on a free kick by Quintero.
The scoreless draw ended a streak of 14 consecutive MLS matches with a goal for the Timbers, a run that dated to October 2021.