NEW YORK (AP) Gerrit Cole dreamed of this moment growing up in California, winning a postseason game in pinstripes at Yankee Stadium.

Harrison Bader had a similar fantasy while sitting in the first base seats as a 15-year-old and watching New York's last championship team.

They combined to make each other's wishes come true.

Cole cruised for most of the night, Bader and Anthony Rizzo homered and the Yankees beat the Cleveland Guardians 4-1 on Tuesday in their AL Division Series opener.

Bader hit his first home run since the Yankees made the surprising deal to acquire him from St. Louis at the trade deadline, tying the score with a third-inning drive to left-center on a Cal Quantrill sinker after Steven Kwan had homered in the top half.

A graduate of Horace Mann School, 5 miles from Yankee Stadium, Bader was in a walking boot with a foot injury at the time of the trade and didn't make his Yankees debut until Sept. 20.

''I want to validate myself,'' Bader said. ''I want to play hard and I want to show my teammates and I want to show -- my parents are in the stands - I want to show them all why I earned that uniform.''

Not even another Josh Donaldson baserunning blunder could slow the Yankees, who have won six straight postseason games against Cleveland dating to a comeback from a 2-0 deficit in the 2017 Division Series. The crowd of 47,807 roared from the first pitch.

''It was just a really awesome experience,'' Cole said. ''Sometimes when you feel the crowd or the energy, it sometimes can become a little easier just to quiet things down because it's so loud. I don't know if that makes sense, but it does to me.''

After Kwan's homer, Cole hit Amed Rosario with a pitch and Jose Ramirez lined a gapper to left-center. Bader, a Gold Glove center fielder, saved a run by cutting off the ball and holding the runners at second and third. Cole escaped a bases-loaded jam by striking out Andres Gimenez with his 60th pitch.

''That may be the at-bat of the game right there,'' Yankees manager Aaron Boone said.

Cole needed just eight pitches in the fourth and allowed just two more runners, on a single and his only walk. He left after giving up one run and four hits in 6 1/3 innings with eight strikeouts.

''We did a really good job early of driving his pitch count up,'' Cleveland manager Terry Francona said. ''And then he had a real quick fourth inning, that kind of got him back into the game.''

Jonathan Loaisiga, Wandy Peralta and Clay Holmes finished with two-hit relief to begin the best-of-five matchup. Holmes hit Owen Miller with a pitch before retiring the final two batters.

''Just had to check on my heart,'' quipped Yankees manager Aaron Boone, who got a pacemaker last year.

AL Central champion Cleveland, playing on the 74th anniversary of its last World Series title, struck out nine times against the AL East-winning Yankees, who returned from a five-day layoff. Guardians batters had the fewest strikeouts in the majors during the regular season.

Jose Trevino put New York ahead with a sacrifce fly in the fifth after Isiah Kiner-Falefa reached on a single into the right-field corner and Wild Card Series star Oscar Gonzalez allowed the ball to bounce through his legs for a two-base error.

''It took a little more spin than I was expecting, and it just went away,'' Gonzalez said through a translator.

Rizzo chased Quantrill with a drive into the right-field second desk in the sixth after Aaron Judge walked.

''The Guardians are a young team that has a lot of momentum and have been pitching really well and playing really well,'' Rizzo said. ''That's a scary combination.''

An inning earlier, Donaldson made another of the lack-of-hustle gaffes marking his first season in New York. With the score tied 1-1, the former AL MVP led off the fifth with a drive that hit the top of the 10-foot wall in right field, just in front of a fan's hands, and bounced back onto the field.

Donaldson went into a trot, thinking it was a home run, and even high-fived first base coach Travis Chapman. But Gonzalez grabbed the ball off the carom and threw to shortstop Amed Rosario, whose throw to first caught Donaldson trying to get back.

''We have to make sure we're getting where we need to get to,'' Boone said.

After the final out, that became a minor matter. Judge, who set an AL record with 62 homers, cited Bader for poor form.

''Judgey told me I ran a little too fast,'' Bader said. ''As soon as he clips it, he knows he's got it. I haven't hit a home run in a long time, so I have to work back into it.''

WEB GEMS

Yankees rookie left fielder Oswaldo Cabrera made a leaping catch against the stand s on Will Brennan leading off the fourth as his cap fell into the hands of a fan. Cabrera threw the ball back to the infield, took a step toward the wall as the fan tossed back his cap, then gave the spectator a high-five with his glove. Two pitches later, Donaldson snagged Austin Hedges' grounder as he slid into foul territory, then got up and made a strong throw to first to retire the slow-footed catcher.

MOUND MATTERS

Quantrill lost for the first time since July 5 following 11 wins over 17 starts. He gave up four runs - three earned - four hits and three walks in five-plus innings.

TRAINER'S ROOM

Guardians: RHP Nick Sandlin (strained shoulder) was dropped along with LHP Kirk McCarty, and RHPs Aaron Civale and Cody Morris were added.

Yankees: INF DJ LeMahieu was left off the roster and may have a broken bone under the second toe of his right foot. ... RHP Scott Effross needed Tommy John surgery.

UP NEXT

After an unusual day off between Games 1 and 2, LHP Nestor Cortes (12-4) starts for the Yankees against RHP Shane Bieber (13-8) on Thursday. Rain is in the forecast.

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