The Sounders (10-5-7, 37 points) won their second straight game and extended their unbeaten streak to four games (3-0-1) before a Major League Soccer season-high crowd of 60,908.
The Galaxy (10-11-3, 33 points) had a two-game winning streak and five-game unbeaten streak (4-0-1) snapped.
Goalkeeper Michael Gspurning picked up his fourth shutout of the season. He played his first home game since May 2 and made his second straight appearance after missing 12 games with a hip injury.
Johnson put the Sounders on the board at the outset of the sixth minute. Mauro Rosales floated a target ball into the penalty area, and Johnson ran onto it, heading it into the back left corner from the top of goalkeeper Josh Saunders' 6-yard box.
Montero scored in the 52nd minute, Caskey hit in the 61st, and Rose finished the scoring in the 88th.
The Galaxy, whose 39 goals are second-most in MLS behind San Jose's 45, were shut out by the Sounders for the second time this season, having dropped a 2-0 decision at Seattle on May 2.