Penalties were needed to settle the 1-1 draw after Crew's Gyasi Zardes struck 11 minutes from the end to cancel out Robin Lod's 18th-minute opener.
Minnesota emerged triumphant on spot-kicks, converting all five of its attempts after Tyler Miller saved Chris Cadden's penalty.
San Jose Earthquakes awaits Minnesota in the quarter-finals.
In the other last-16 fixture, the Timbers were 4-2 winners on penalties against Jaap Stam's FC Cincinnati.
Jaroslaw Niezgoda broke the deadlock for the Timbers when he coolly found the back of the net from inside the six-yard box after team-mate Sebastian Blanco was left unmarked before sliding a pass to the Polish forward.
Cincinnati restored parity after Jurgen Locadia converted an 81st-minute penalty to force a shoot-out.
Portland was not to be denied as Niezgoda scored the decisive penalty after Cincinnati's Kendall Waston missed and Locadia had his attempt saved.
The Timbers will go head-to-head with New York City for a place in the semi-finals.