The visitors were the first to threaten as Jose Villarreal came close to finding a breakthrough when he fired at goal from outside the box, but Steve Clark was there to make the save.
Los Angeles took the lead when Robbie Rogers threaded a through ball into the path of Sebastian Lletget, and the midfielder managed to fire the ball through Clark from a tight angle.
Just before the break, Lletget almost doubled his tally for the evening when he latched onto a loose ball and drilled a strike at goal, but Clark made a solid stop.
In the second half, Crew responded and carved out an opportunity as Hector Jimenez lifted a cross into the box for the run of Justin Meram, who put his header wide of the target.
Columbus found their equaliser in the 66th minute as Ethan Finlay found Federico Higuain on the edge of the box, and the midfielder curled his effort beyond the reach of Jaime Penedo into the top corner.
The home side pushed forward in search of the lead and Meram almost handed them the advantage as he powered Finlay's cross at goal with a header, but Penedo tipped the ball over the bar.
Both teams pushed for a winner, but were unable to find another breakthrough as they settled for a point apiece.