The Yankees' Aaron Judge, right, and Juan Soto celebrate after...

The Yankees' Aaron Judge, right, and Juan Soto celebrate after scoring runs during the sixth inning of a game against the Seattle Mariners on Tuesday in Seattle. Credit: AP/Stephen Brashear

SEATTLE — Promising Mariners righthander Bryan Woo had his way with the Yankees the first two times he faced them in his young career.

The Yankees, Aaron Judge and Juan Soto in particular, had their way with the 24-year-old Tuesday night.

With Judge driving in four runs in the first two innings — upping his MLB-leading RBI total to 136 — and Soto adding a double, a walk and a two-run homer against Woo in his first three times to the plate, the Yankees coasted to an 11-2 victory over the still playoff-hopeful Mariners in front of 31,668 at T-Mobile Park, cutting their magic number for clinching a postseason bid to one.

The 12-hit outburst, which also included three hits by Gleyber Torres, a home run by Jasson Dominguez and a three-run double by AL Rookie of the Year candidate Austin Wells, helped the Yankees (88-63) up their lead in the American League East to four games over the Orioles, who were blown out at home by the Giants.

A victory by the Yankees on Wednesday night officially qualifies them for October baseball.

“We’ll definitely be excited, especially after the season we had last year, missing out on the postseason,” said Judge, whose two-run double in the first off Woo made it 2-0 and whose two-run single in the second made it 4-0. “That’s what we came into the season to do, get into the postseason and give ourselves an opportunity to go out there and win a World Series. That’ll be step one. But we gotta get there first.”

Soto’s two-run homer, a two-run opposite-field blast in the fourth, made it 6-1 and simultaneously achieved three milestones for the 25-year-old rightfielder: It gave Soto his first career 40-homer season; gave him his 200th career homer; and allowed him to cross a career goal off his list — homering in every ballpark. The latter, Soto said afterward, was the most significant of the three.

“It’s a great feeling for me,” said Soto, who mentioned it as a goal back in the spring. “So many ballparks, so many different dimensions. I just wanted to get all of them checked [off]. What a way to go to free agency, with homers in all 30 ballparks.”

Dominguez’s fifth-inning homer made it 7-1 and Wells made it 10-1 in the sixth with his double.

The blowout allowed Marcus Stroman, shifted to the bullpen last week, to make his 2024 debut as a reliever. The righthander allowed two hits in a scoreless seventh and a solo home run to Luke Raley in the eighth, which made it 10-2. Stroman finished the ninth for his second career save (the other came in 2014).

Oswaldo Cabrera’s RBI single in the ninth off Leo Rivas, a position player throwing pitches in the mid-40-mph range, made it 11-2.

Woo came in 8-2 with a 2.38 ERA this season and having not allowed a run in 11 1/3 career innings against the Yankees (including six scoreless in a 6-3 victory May 21 at the Stadium). But it took the Yankees all of nine pitches to take a 2-0 lead.

Luis Gil, though facing traffic throughout, mostly because of command issues, still made it through five innings, allowing one run and four hits in improving to 14-6, 3.14 and doing nothing to damage his hopes of landing a spot in the Yankees' postseason rotation.

An odd 24-pitch bottom of the first started with Gil hitting leadoff man Victor Robles on the right hand with his first pitch of the night and ended with Robles, who left the game in the third with a right hand contusion, making the inning’s final out at the plate with an ill-advised attempted steal of home.

Judge then made the pitcher’s night even easier with his single that provided a four-run cushion.

“He really set the tone for us tonight with his big two first at-bats,” Aaron Boone said of Judge.

At this time last season under Boone, the Yankees were staggering to an 82-80 record and fourth-place finish in the East. As of late Tuesday night, they were a virtual lock to make the playoffs, pulling away in their pursuit of the East crown, and headed for an apparent photo finish with the Guardians for the AL’s best record.

“We’re better,” Boone said. “We weren’t a great team last year, and this team has a chance to do something special…We haven’t done anything yet, but so far we’ve put ourselves in a position to go achieve that.”