The Rapids will move on to meet Kansas City in a two-match, aggregate score showdown that begins in Colorado Sunday.
This is the second year in a row Colorado has eliminated Columbus from the playoffs, doing so last year in the conference semifinals.
The game's only goal came when Brian Mullan sent a pass to Kosuke Kimura near the end line. Kimura fed it back to Cummings in the box and he drove a right-footed effort past goalkeeper William Hesmer.
It was the first playoff goal in Cummings' career.
Colorado joined New York in moving on from the wild-card round in the playoffs. New York defeated Dallas 2-0 Wednesday.