RSL (6-0-5) hasn't dropped a game in its first 11 matches this season and is the only undefeated team in Major League Soccer.
Plata scored in the 23rd minute with a roller past goalkeeper Clint Irwin to the far post.
Javier Morales scored on a penalty kick in the 52nd minute to put RSL up 2-0. It was Morales' fourth goal.
Salt Lake was without starting goalkeeper Nick Rimando and captain Kyle Beckerman, who were among 30 players training at Stanford for a spot on the U.S. men's team for the World Cup this summer. Forward Alvaro Saborio was away with the Costa Rican national team.
The Rapids (4-4-3) avoided the shutout with defender Drew Moor's header into the corner in the 83rd minute.
Salt Lake is coming off a 5-2 victory over the Houston Dynamo last weekend, while the Rapids lost 3-1 at home to Chivas USA.
The Rocky Mountain Cup was created in 2005, Real Salt Lake's first season in MLS.
Last season the Rapids won the rivalry with a win and two draws. Real Salt Lake leads the all-time competition 6-3.
The home team has not lost in Rocky Mountain Cup play for the last two seasons.
RSL will try to extend its season-long unbeaten streak to 12 Saturday night at home against FC Dallas.
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