The Dynamo were presented with a great chance to take the lead when Will Bruin pounced on a mistake by Will Trapp and raced through on goal, but his shot was smartly saved by Steve Clark.
Columbus missed a glorious opportunity when Adam Bedell knocked the ball into the path of Federico Higuain two yards from the net, but he somehow put his header over the bar.
The miss did not haunt Higuain for long as his side took the lead as Justin Meram dispossessed David Horst on the edge of his own box and weaved his way into the area before firing past Tally Hall.
Horst had a chance to get his side level just before the break as he beat the offside trap from a free-kick delivery into the box, but he got his attempt all wrong as he headed well wide of the target.
In the second half, the Crew doubled their lead as Meram whipped a dangerous ball into the box and Bedell outmuscled Horst in the area to guide a headed effort past Hall.