The Loons will play the San Jose Earthquakes on Saturday.
Minnesota goalkeeper Tyler Miller stopped Chris Cadden of the Crew in the bottom of the third round to preserve a 3-2 advantage. After Raheem Edwards of Minnesota and the Crew's Lucas Zelarayan converted, Chase Gasper scored past Andrew Tarbell for the insurmountable lead.
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The tournament winner earns a spot in the 2021 CONCACAF Champions League.
Minnesota took the lead in the 18th minute on a goal by Robin Lod. Gyasi Zardes tied it in the 79th for the Crew after he scored on his own rebound of a penalty kick save by Miller for his fifth goal of the season.
The penalty was awarded when Jose Aja fouled Derrick Etienne Jr. in the corner of the box. Miller made a diving stop of Zardes' attempt, but the ball went directly to the forward for an unmarked follow-up.
Miller stopped a Zelarayan shot in the closing seconds to send the match to PKs.
The Crew won all three Group E matches by a combined 7-0 score but quickly faced their first deficit this season when they couldn't defend a corner kick that was drawn by former Columbus midfielder Ethan Finlay.
Columbus cleared the first ball, but it bounced around to Lod for his second goal. It was the first score conceded by the Crew in 299 minutes.
Minnesota midfielder Kevin Molino (hamstring) was expected to be available, but he did not play.
The knockout stage matches at the ESPN Wide World of Sports Complex do no count as part of the regular-season standings though the group-stage matches did.