NEW YORK -- The ovation came in loud and strong, rolling in waves through Yankee Stadium’s faraway decks and accompanying Gerrit Cole for his journey toward the first-base dugout. After three years in pinstripes, finally hearing the cheers of a home playoff crowd, he reached for the bill of his cap and doffed it ever-so-slightly. He had done his job, and done it well.

In the moment, Cole played his seventh-inning departure cool, partially because he’d left some traffic on the basepaths with eight outs still separating his club from victory in Game 1 of the American League Division Series. Only after the Yankees had secured their 4-1 win over the Guardians on Tuesday did Cole’s steely gaze melt, an appreciation of having met the moment.

“It was very special for me,” Cole said. “The game’s not over, so it’s not the most comfortable time to acknowledge the crowd, but I certainly felt it and appreciated it. I thought they were in every pitch tonight. What a wonderful experience to have them behind us.”

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On the winter morning he set pen to paper, agreeing to the largest contract ever issued to a free-agent pitcher, Cole could never have anticipated it might take this long to experience the elation of a Bronx postseason crowd. The pandemic-shortened 2020 season and an all-too-brief ’21 playoff run kept Cole from living that boyhood dream; now he’s hungry for more.

“It was just a really awesome experience overall,” Cole said. “What a great atmosphere. And it was my dad’s birthday today, so it was a good day.”

Cole largely disregarded any chatter about ceding his Game 1 start to another starter, like Nestor Cortes or Luis Severino, though there had been some discussion behind closed doors about the identity of the best choice. Cole set a franchise record with 257 strikeouts during the regular season, but also led the AL with 33 homers allowed, prompting concern.

To Cole, who has spoken openly about relishing the high stakes of the postseason, there should have been no debate. The ace delivered on the big stage, scattering four hits and striking out eight. In five postseason starts as a Yankee, Cole is 3-1 with a 3.38 ERA and 41 strikeouts, stats that include last year’s abbreviated outing in the AL Wild Card Game at Fenway Park.

“Gerrit was really good,” manager Aaron Boone said. “He did a really good job of owning the moment and being unpredictable. And I thought his breaking ball was really on point.”

Harrison Bader and Anthony Rizzo homered in the win for the Yankees, who won five of six from the Guardians during the regular season, outscoring Cleveland by a 38-14 margin.

“It feels really good to come out and get Game 1,” Rizzo said.

The evening got off to a shaky start for Cole, with Steven Kwan’s third-inning home run sparking a string of heavy traffic on the basepaths. A hit-by-pitch and double created trouble as Josh Naylor reached on a fielder’s choice, with the first baseman Rizzo firing home to catcher Jose Trevino as Amed Rosario scampered back to third base.

There would be no repeat of Cole’s Fenway playoff disaster. Cole induced a fielder’s choice, then struck out Andrés Giménez.

“That was Gerrit’s time to pick us up,” Rizzo said. “You saw him bear down even more and get out of that inning. That was big for us. Hopefully that’s going to be the theme throughout October, us picking each other up.”

Bader homered to left field off Cal Quantrill in the home half of the third, marking the Bronxville, N.Y., product’s first long ball in pinstripes. Cole enjoyed an eight-pitch fourth inning and began to cruise, taking stress off an injury-depleted bullpen.

“I thought we did a really good job early of driving his pitch count up,” Guardians manager Terry Francona said. “Then he had a real quick fourth inning. That got him back into the game.”

Said Trevino: “His stuff was working tonight. He was confident in everything he threw. I thought we did a good job of mixing pitches, mixing locations.”

New York grabbed the lead in the fifth; Josh Donaldson thought he’d homered to the short porch in right field, but the ball hit near the top of the fence and ricocheted back into play, with Donaldson tagged out at first base.

No matter; Isiah Kiner-Falefa followed with a single, advancing to third on an error by right fielder Oscar Gonzalez, then scoring on a Trevino sacrifice fly. Rizzo extended the lead in the sixth with a two-run homer to the second deck in right field, knocking out Quantrill, who had not lost since July 5 -- a span of 17 regular-season starts.

Cole’s 101st pitch was his last of the evening, whacked by Myles Straw for a liner that tipped Kiner-Falefa’s glove for a one-out single. Cole watched from the dugout as Wandy Peralta iced the rally, inducing an inning-ending double play, then applauded when Clay Holmes polished off the ninth inning for his first postseason save.

“A good day’s work, and get ready for the next one,” Cole said. “It’s kind of simple.”