Aaron Judge follows through on his solo home run against...

Aaron Judge follows through on his solo home run against the Toronto Blue Jays during the fourth inning. Sept. 30, 2017. Credit: Kathleen Malone-Van Dyke

The Yankees just ran out of time in the end. They won game after game of this homestand, threw their best available pitcher at the Blue Jays Saturday, and never quite let up on the small hope that they could make good on a goal that few thought they could ever achieve even a few short months ago.

Problem was, even though the Yankees defeated the Blue Jays, 2-1, at the Stadium Saturday, the Red Sox won, too. The Yankees’ mad dash for the AL East title was officially over a couple minutes after “Start Spreading the News” had stopped playing at Yankee Stadium. Instead, they’ll host the Twins Tuesday in a wild-card elimination game.

It’s hardly ideal — manager Joe Girardi has conceded as much, and showed how desperately he wanted the division Saturday morning, when he decided to pitch CC Sabathia over the scheduled starter, Jaime Garcia. But he also noted that he couldn’t be disappointed in how his team played in this final stretch, a performance, he said, that will serve them well when they fight for their baseball life next week. The Yankees enter the final game of the regular season having gone 5-1 so far on this homestand, making up ground on the shaky Red Sox at a quick pace.

“As close as I’ve ever watched,” said Girardi, when asked if he snuck any glances at the scoreboard (the Red Sox won, 6-3).

“In a sense [these last few games were like] elimination games for us,” Girardi said, adding that this was a good thing for a team full of youth and postseason inexperience. “We were five down coming home from the road trip and our guys just continued to play well. They continued to play loose. They continued to make pitches, grind out at-bats and that was encouraging to me because it’s not like we’ve had a chance to turn it off and I like that.”

Luis Severino is set to face Ervin Santana in the elimination game, but by Saturday, the nip in the air and the on-field product already made it already feel like October baseball.

“The chemistry,” said Sabathia, when asked about what distinguishes this team. “This is a great clubhouse. One of the best I’ve been in in a long time, so I think we’re on the right track to put a run together.”

Sabathia was stellar, and Aaron Judge dropped more than a few jaws with his monster home run in the fourth — one that, according to Statcast, measured 484 feet long, making it the fourth-longest home run hit in the majors this year.

Judge’s leadoff homer of Marcus Stroman gave the Yankees a 1-0 lead in the fourth, and then they tacked on another — making it the only support Sabathia would get. Didi Gregorius had a one-out single to left, moved to second on a wild pitch and then came home on Starlin Castro’s single up the middle.

The Blue Jays cut the lead in half in the eighth, when Josh Donaldson scored Ezequiel Carrera on a sacrifice fly. But David Robertson — who replaced Tommy Kahnle with two on and no outs — struck out Justin Smoak with the tying run at second to end the inning.

Sabathia did everything he was asked and more. He allowed no runs on four hits in 5 2/3 innings, and faced the minimum through four. He was the beneficiary of two double plays, and the Blue Jays only threatened against him in the sixth, when Ryan Goins hit a ball to deep right that was snagged by a fan a few inches from the lip of the wall. The play was ruled fan interference and Goins was awarded a double. Sabathia threw 75 pitches, struck out six, walked none, and was pulled for Chad Green after the Red Sox had staked a substantial lead over the Astros, essentially ending the Yankees’ division hopes.

At 37, Sabathia (14-5) ended the regular season with a 3.69 ERA — his lowest season ERA since his last All-Star season, in 2012.

“An outstanding performance,” Girardi said. “We’ve seen that from CC since we got him, I think nine years ago. The determination in him, the competitiveness in him. He’s somewhat reinvented himself.”

He has, and if this season is any indication, the Yankees have, too.

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