NEW YORK — With the season on the line, Masahiro Tanaka delivered what he called his most important win as a Major Leaguer and Greg Bird cracked a majestic home run off Andrew Miller, extending the Yankees’ hopes for another day with a 1-0 victory over the Indians in Sunday’s Game 3 of the American League Division Series presented by Doosan.

Bird’s second homer of the series, a towering drive into the second deck in right field, provided some desperately needed support behind Tanaka’s dominant outing. With the Indians flailing regularly at his lethal splitter, Tanaka blanked Cleveland’s formidable offense on three hits through seven frames, issuing one walk with seven strikeouts over a 92-pitch effort.

“I came here to pitch in these types of games, and to be able to help the team win in these types of games,” Tanaka said through an interpreter. “As a player, those are the moments that you want to go in there and shine the most.”

New York trails the best-of-five series, 2-1, with Game 4 set to be played at Yankee Stadium on Monday at 7 p.m. ET. It marked the Yankees’ first 1-0 postseason victory since Game 3 of the 2001 ALDS against the Athletics, a series in which the Yanks rallied from an 0-2 deficit.

Offering the Yankees their longest start of the postseason, Tanaka had some help from his defense. Aaron Judge needed every inch of his 6-foot-7 frame to corral a deep Francisco Lindor drive in the sixth inning, taking a two-run homer away from the Tribe shortstop and drawing chants of “M-V-P!” from the crowd of 48,614.

“A great catch, ends up saving the game,” Yanks manager Joe Girardi said. “I’ve said all along about Aaron Judge, he’s a complete player. It’s not just a home run hitter. It’s a guy that runs the bases, that plays very good defense, and that drives in a ton of runs and is extremely productive as a hitter.”

Off the bat, Lindor did not think his deep fly would carry as far as it did.

“Not at all. I just hit it off the end,” Lindor said. “It was a pitch low and away. I didn’t think I had it. For a second, I was like, ‘Aw, it’s popped up.’ But then I saw it was carrying. That’s when you start hoping, you know, you’re like, ‘Please, go, go.’ But Judge made a good play on it.”

Indians starter Carlos Carrasco limited the Yankees to three hits over 5 2/3 scoreless innings in the first postseason start of his career, walking three and striking out seven. Carrasco matched Tanaka into the sixth, when Tribe manager Terry Francona called upon Miller to retire Starlin Castro on a popout that left the bases loaded.

“I thought he was tremendous,” Francona said of Carrasco. “He came out, looked like he was using the adrenaline to his advantage, throwing a nice, crisp fastball and real good offspeed. That’s two of the better starting performers you’re going to see, because Tanaka was, unfortunately, that was one of the better games we’ve seen all year.”

The Yanks used David Robertson for just one out in the eighth before calling upon closer Aroldis Chapman for a five-out save. Chapman pitched around a pair of singles in the ninth, retiring Carlos Santana on a flyout to center for the final out. Chapman recorded five or more outs just twice during the regular season, including a Sept. 18 save against the Twins.

“This is a decisive game. You can’t hold back,” Chapman said through an interpreter. “Everything you have, you have to go out there and give it all. WIthout tonight, there’s no tomorrow.”