Smith suffered the injury in the 72nd minute without contact, and will be out at least six months. Colorado opened the playoffs with a 1-0 victory over the Columbus Crew at Dick's Sporting Goods Park in the wild-card round.
The 30-year-old Smith will have surgery on Tuesday, and leave the Rapids void of another key player for the rest of the season. Striker Conor Casey ruptured his Achilles earlier this season.
Smith appeared in 20 regular season matches for Colorado. With him sidelined, coach Gary Smith will have a headache in midfield for the Eastern Conference semifinal series against Sporting Kansas City.
Captain Pablo Mastroeni was sidelined after a concussion, and Smith replaced him in the lineup. Mastroeni will still be sidelined for the first leg Sunday in Colorado, and his status for the second leg is unknown.