Dillon Powers and Marcelo Sarvas scored in the final 13 minutes of the clash to leave Sporting KC needing a home win over Los Angeles Galaxy on Sunday to put them into the playoffs.
The Rapids came close to taking the lead within the first 30 minutes of the match when Kevin Doyle's diving header looked goal-bound, but goalkeeper Tim Melia was able to keep it out.
A poor finish from Luis Solignac, who sent a decent chance high over the crossbar, kept proceedings level at Sporting Park before the break, and the home side failed to strike in a first half for the ninth league game in a row.
The home side upped their game as the half went on, but they struggled to make the most of their possession in the final third, as Zusi, Soni Mustivar and Krisztian Nemeth all missed chances.
The Rapids were able to take advantage of their opponents' attacking approach by hitting them on the counter, and in the closing stages of the match, Doyle and Powers caught Kevin Ellis and Mustivar out of position.
Doyle laid the ball off to Powers, who chipped over the goalkeeper and found the back of the net to put the visitors 1-0 up with just 13 minutes of normal time remaining.
Sporting KC almost equalised in the 82nd minute when Jordi Quintilla came close, but his effort flew just wide of the target, and two minutes later the hosts were dealt a double blow when Sam Cronin was sent off for a second yellow.
The Rapids took advantage again on the break as Sarvas grabbed his side's second to leave Sporting KC fifth in the Western Conference, while the visitors reside bottom of the table.