Ruidiaz gave Sounders the lead in the 23rd minute from the penalty spot after Erick Avalos had fouled Cristian Roldan.
Montero (64) and Lodeiro (70) then netted two quick goals in the second half to put the game beyond any doubt.
Sounders coach Brian Schmetzer could have rotated his roster for the game, but opted to name a strong lineup.
"Our team, I didn't have to say anything before the game. We are blessed to have great players," Schmetzer said after the match. "We're a humble club, we work, we try to achieve great things and we're always trying to win. That is part of why our decision to put a strong lineup out there tonight against a very good Tigres team, it was important for us, it was important for this club.
"It's important for us to showcase the players that we have."
Sounders will now play Orlando City or Santos Laguna in the semifinals next month.