The Crew had an excellent chance to take the lead in the 15th minute when a loose ball from a corner fell to Chad Barson inside the six-yard box, but he scuffed his left-footed shot well wide of the post.
Montreal responded and had an excellent chance of their own as Justin Mapp found space inside the box and poked his low shot just wide of the net.
Marco di Vaio was presented with a decent chance to fire the visitors ahead when he did well on the edge of the box to create room for a drive, but his shot had just too much power to trouble Steve Clark.
However, the Impact did take the lead in the 35th minute when Di Vaio this time managed to find the net as he turned Giancarlo Gonzalez one way then the other before drilling a shot into the bottom corner.
The Crew were indebted to Clark for keeping them level when Di Vaio raided into the box and saw his shot well saved, while Andres Romero saw his effort on the rebound tipped past the post.
Columbus nearly used those saves to their advantage as they came within inches of taking the lead as Aaron Schoenfeld saw his glancing header clip the outside of the post from Federico Higuain's corner.
The home side did get their winner when Anor picked up the ball in the centre circle and drove at the Impact backline until he reached the edge of the box, where he unleashed a powerful shot which found the top corner.