It was Nyassi's first three goals of the year and only the seventh time a Rapids player had scored three times in a game. Wells Thompson added a goal for Colorado (7-6-9).
The Rapids needed a scoring lift as leading scorer Conor Casey earlier in the day had surgery to repair a torn Achilles tendon. Casey, the last Rapids to have a three-goal game, is expected to be sidelined for at least eight months.
Thierry Henry scored his 10th goal of the year for New York (6-5-11), who had been blanked in its previous two games.
The Red Bulls are winless, 0-2-7 in their last nine road games.