Colorado continued to threaten in the first half and the visitors were grateful to defender Hector Jimenez for an interception on the line as he stopped Marvell Wynne's cross from finding Sanchez at the back post.
Columbus then had an opportunity of their own when Ben Speas lifted a cross into the area for Justin Meram, who directed a header towards the bottom corner only to see his effort beaten away by Clinton Irwin.
After the break, the Crew came close again when they won a free kick just outside the penalty area, but Bernardo Anor's effort went inches over the bar.
However, the Rapids took the lead as Sanchez collected Wynne's cross from the right and managed to head the ball in the back of the net, with the aid of a deflection off Eric Gehrig.
Columbus responded and almost drew level as Jairo Arrieta found space through the middle as he latched on to a through-ball but his low shot was well saved by Irwin, who was quick on his feet to gather the rebound.
Irwin could do nothing to prevent the Crew finally finding the net as the visitors capitalised on two errors from Dillon Powers and Jared Watts, which allowed Ethan Finlay to earn a point for his side as he blasted the ball into the top corner.