The result sets up a semifinal clash with St. Louis CITY SC, who defeated Houston Dynamo FC in penalty kicks on Tuesday.
Colorado opened the scoring in the 40th minute, as Yapi finished off a corner-kick routine with a close-range header at the far post, assisted by Dante Sealy.
The Rapids doubled the advantage in first-half stoppage time, with Navarro converting from the spot after a handball in the penalty area on San Jose’s Reid Roberts.
Colorado will host St. Louis in the semifinals as they seek their first-ever US Open Cup title, with their lone previous appearance in the tournament final coming in 1999.