The Sounders, who led the Western Conference for long periods of the season and let top spot slip on the final day, went down 6-5 on penalties after a goalless draw where RSL did not have a shot on goal across 120 minutes on Tuesday.
RSL goalkeeper David Ochoa was the hero, denying Kelyn Rowe with the first spotkick in sudden death, before Justen Glad's effort went in off the post to clinch their progress to a Western Conference semi-final against Sporting Kansas City.
Seattle appeared on their way to victory in the shootout when Stefan Frei saved from the second kick from Pablo Ruiz but it was re-taken, and converted, due to the goalkeeper coming off his line prematurely.
The Sounders had come closest to scoring during regulation time, with Raul Ruidiaz's second-half effort hitting the crossbar.
Hany Mukhtar's double earned second-year franchise Nashville a 3-1 come-from-behind victory over Orlando City in their MLS Cup first round clash.
US international Daryl Dike had headed Orlando into a 14th-minute lead, but Nashville equalised seven minutes later via Mukhtar's deflected strike from just outside the box.
Mukhtar put Nashville ahead in the 74th minute after a spectacular run into the box before unleashing a low left-foot shot which beat Orlando goalkeeper Pedro Gallese.
Jhonder Cadiz added another deep into stoppage time to seal Nashville's passage into the Eastern Conference semi-finals where they will face Philadelphia Union.