Dayton found Gallegos in open space and delivered a pass to score the game’s first goal in the 10th minute.
Ruiz scored a few minutes later and then converted a second-half penalty to secure the victory.
The No. 1 seed Valley Christian (9-7-4) finished fifth in the tightly contested Western Catholic Athletic League and was the last team to beat league champion St. Francis.
Woodside ranked fifth in the Bay Division of the Peninsula Athletic League and had an overall winning percentage of .579 (7-2-0 outside the league), third best among teams in the PAL Bay Division.
The Wildcats have gone 0-3-1 in their last four regular season games. They opened the season with wins in six of their first eight matches.
Woodside lost their first round match in last year’s CCS Division I playoffs at home to Watsonville after winning the PAL Ocean Division title.
Menlo-Atherton (7-6-4) also picked up a big win by beating host Menlo School 2-0 in Division II.
Top-seeded Mountain View (19-0-1) beat Burlingame 4-0 in the first round of the CCS Open Division playoffs.
Karena Shah scored twice and registered an assist for the Spartans, who will host St. Ignatius or Leigh in Tuesday’s semi-final.
Seventh-seeded Menlo School hit the road at No. 2 Live Oak and held a scoring clinic, returning to Atherton with a 6-0 win on Saturday in the CCS Division II game.
The Knights (11-5-4) will also be on the road for Wednesday’s semifinal against No. 3 Christopher, a 2-1 winner over Menlo-Atherton (6-7-6).
Christopher (10-5-1) took a 1-0 lead from a corner kick seconds before halftime. The Bears tied the game with a penalty midway through the second half, but Christopher scored again in the final minute to claim the win.
Christopher’s keeper saved the game for her team, snatching some goals from MA once when she came out of the net and blocked a shot with her body at the 15-yard line.
Also in Division II, host Sacred Heart Prep edged Aptos 3-2 and Palo Alto fell on penalties to top seed Piedmont Hills, 4-3. The game was tied 1-1 in regulation time.
Sacred Heart Prep (12-6-3) visits Piedmont Hills (17-2-2) on Wednesday.
In Division I, No. 8 Gunn fell to No. 1 Carlmont, 1-0, and No. 6 Sequoia lost to No. 3 Saratoga, 2-1.
Will be updated.