The Yankees wanted the Red Sox as a playoff opponent. Things didn’t go as well as they thought they might

Kyle Schwarber (left) gave the Red Sox a lift with a solo homer in the third inning.Barry Chin/Globe Staff

Amid the dizzying discussions of postseason scenarios that hovered over the final days of the regular season, a particularly intriguing possibility took shape.

Had there been a four-way tie among the Red Sox, Yankees, Blue Jays, and Mariners, the Sox and Jays would have been awarded hosting duties for two Games 163. The Yankees, by virtue of a better record than Seattle, had the choice of whether they’d prefer to play on the road in Boston or Toronto. Early on Sunday, the Yankees chose the Red Sox as their preferred opponent.

The four-way tie didn’t materialize. The Sox and Yankees won on Sunday, clinching outright wild-card berths and a meeting at Fenway. But it wasn’t lost on the Sox that the matchup was one the Yankees had identified as their postseason preference.

“We knew about it,” Sox center fielder Kiké Hernández said on Tuesday afternoon. “They wanted us and they got us now. So, win or go home.”

On Tuesday, the Yankees went home, while the Red Sox — in the first win-or-go-home Wild Card Game in franchise history — delivered a dominating 6-2 win to advance to the Division Series against Tampa Bay. The victory represented the first home postseason clincher for the Sox since 2013, and improved them to 8-1 against the Yankees in postseason meetings dating to the 2004 ALCS.

Fenway surged with its most electric atmosphere in years. Player introductions of the rivals brought impassioned reactions that elevated further when NESN analyst Jerry Remy threw out the first pitch to longtime teammate and broadcast partner Dennis Eckersley.

The crowd of 38,324 remained standing through the first inning, turning raucous when Nate Eovaldi escaped — after a two-out single smashed off the Wall by Giancarlo Stanton — by striking out Joey Gallo on a nosediving splitter.

After Yankees starter Gerrit Cole — struggling with his command out of the gate — issued a two-out walk, he fell behind Xander Bogaerts, 2 and 1, on a curveball and two sliders.

Untrusting of his fastball, Cole attempted a right-on-right changeup, a pitch he seldom uses but that tends to be effective (.118 average, .148 slugging percentage by righties) when he does. But in this instance, the pitch sat squarely in the middle of the strike zone.

A two-run homer in the first inning from Xander Bogaerts set the tone for the evening.Jim Davis/Globe Staff

The Red Sox shortstop sent a 427-foot rocket into the center-field bleachers for a 2-0 lead. His third career postseason homer brought the crowd to a crescendo that did not abate against the rattled Cole.

“For the guy who’s been here through multiple championships to step up and do it tonight, you can’t say enough good things about him,” said Kyle Schwarber of Bogaerts.

Though Cole worked around a Kevin Plawecki double in the second, the Sox looked comfortable against all of his pitches by the time they started their second tour through the order in the third.

Schwarber, pressed into leadoff duty by a lineup overhaul necessitated by the loss of J.D. Martinez, obliterated an elevated 97-mile-per-hour fastball from Cole. His seventh career postseason homer sailed over the Yankees’ bullpen, crashing in the right-field bleachers an estimated 435 feet later, giving the Sox a 4-1 lead.

When Hernández followed with an infield single and Cole walked Rafael Devers for the second time, Yankees manager Aaron Boone employed a stunningly swift hook of his ace. In 13 postseason outings entering the Wild Card Game, Cole had never thrown fewer than five innings. On Tuesday, he lasted just two, tied with a pair of clunkers as a Pirate in 2016 for the shortest start of his career.

“For us to get him out that early, it was a huge win for us,” said Schwarber.

Reliever Clay Holmes staved off a blowout by striking out Bogaerts and getting Alex Verdugo to ground into a double play. Still, the three-run advantage seemed sizable given the work of the Sox starter.

Nathan Eovaldi was dominant Tuesday night against New York.Stan Grossfeld/Globe Staff

Eovaldi, the most dominant Red Sox force of the 2018 postseason, once again looked the part of an October difference-maker. On a night when he was throwing harder than at any point this year, averaging a season-high 98.0 m.p.h. on his four-seam fastball, Eovaldi also showed feel for his other four pitches. The result was dominance, with Eovaldi cruising through five scoreless innings with seven strikeouts.

“Nate was amazing,” said manager Alex Cora.

But after he fanned pinch hitter Rougned Odor to open the sixth, the third time through the order began inauspiciously. Leadoff hitter Anthony Rizzo sent a towering solo homer inside the Pesky Pole to put the Yankees on the board, and Aaron Judge followed by tapping an infield single.

In another era, Eovaldi, who was at just 71 pitches, would have continued. But this is the era of Snell-ing, in which the manager’s responsibility is to pull a starter before potential disaster rather than in response to an actual one.

And so, Cora called for Ryan Brasier to face Stanton.

“He looked at me like, ‘What are you doing?’ ” said Cora of Eovaldi.

The move almost catalyzed catastrophe, as Stanton nearly knocked down the Wall in left-center with a 115-m.p.h. rocket.

Yet the Red Sox benefited from the ferocity of the blast. Yankees third base coach Phil Nevin waved Judge to the plate. But the ricochet off the Wall came so quickly and forcefully that, though the ball careened past left fielder Verdugo, Hernández — alertly backing up — had time to make a one-hop throw to the infield, where Bogaerts patiently relayed the ball to the plate to cut down Judge.

“Play of the game,” said Schwarber.

Kevin Plawecki tags Aaron Judge out at the plate in the sixth inning, one of the biggest plays of the game.Jim Davis/Globe Staff

Gifted an out, Brasier ended the sixth by retiring Gallo on a popup. The Sox quickly restored their three-run lead in the bottom of the inning, as Verdugo drilled a double to right off Luis Severino to score Bogaerts and put the Sox ahead, 4-1.

One inning later, after a scoreless inning from Tanner Houck, Verdugo broke the game open, lining a two-out, two-run single to left-center to push the lead to 6-1. From there, a scoreless eighth from Hansel Robles and a one-run ninth by rookie Garrett Whitlock, a former Yankees prospect whom the Sox selected from New York in the Rule 5 draft last December, represented game-ending formalities.

On Thursday, the Sox will open their ALDS against the Rays, one of just eight teams remaining in the playoff tournament. The Yankees will start contemplating a dissatisfying end and a winter of change.

“Sometimes it looks horrible, but 93 times this year it hasn’t looked horrible, so we’re going to keep rolling,” said Cora. “Now we go to the next one, and we’ve just got to be ready to face a great baseball team … We have a huge challenge. But we’re ready for it.”