Gehrig came within inches of heading his side ahead minutes later when he latched onto Higuain's corner, but his effort struck the post and bounced clear.
In the second half, the Crew took the lead as Higuain curled a free kick from distance into the top corner past the outstretched hand of Shuttleworth.
Nguyen had the opportunity to respond from a set piece for the visitors, but he saw his effort well saved by Clark at the second attempt.
Columbus were given the chance to effectively seal the game when they were awarded a penalty after AJ Soares fouled Justin Meram in the area in the 83rd minute. However, Shuttleworth denied Higuain his second goal of the game with a solid stop.
Charlie Davies almost made Higuain pay for his error when he raced through on goal, but Clark produced a fingertip stop to turn his shot past the post, which was enough to secure the home side the three points.