Bucky Dent was in the stadium, the calendar said October, and each team needed to win to keep its season alive. It had been 43 years since Dent’s famous home run helped the Yankees win the last elimination game between these storied rivals at Fenway Park, and nearly a half-century later, Boston finally got a small measure of revenge for that particular game.

This time Xander Bogaerts and Kyle Schwarber hit home runs off Gerrit Cole, as the Red Sox beat the Yankees, 6-2, in the American League wild-card game. Boston advance to an A.L. division series against the Tampa Bay Rays, the best team in the A.L. during the regular season.

The Yankees and Red Sox both contended for the A.L. East title this year, but neither could outlast Tampa Bay, and their fumbling away of opportunities over the last few weeks meant they would face each other in Tuesday’s one-game playoff, two imperfect teams hoping to fight on.

In 1978, they were both powerhouses.

Boston has shifted the nature of the rivalry in the 21st century, exacting its real revenge for 1978 by stunning the Yankees in the 2004 American League Championship Series. But for older Red Sox fans, there was still some unfinished business. The memory of Dent’s home run still stings, and Tuesday’s game perhaps provided some long-awaited salve to the ancient wound.

Cole, the Yankees’ nominal ace who has had an uneven season, did not make it out of the third inning and will probably hear plenty about it into next year. He signed a nine-year, $324 million contract before the 2020 season, in part because he was expected to win this kind of game. But he gave up a two-run home run to Bogaerts in the first inning and a sky-scraping bases-empty shot to Schwarber in the third.

Two batters later, Cole walked Rafael Devers, and Aaron Boone, the Yankees’ manager, walked from the dugout and took him out. As Cole watched nervously from the bench, reliever Clay Holmes escaped the jam, with two men on base, by striking out Bogaerts and inducing a double-play ball from Alex Verdugo.

Cole’s season has been defined by a combination of controversy and inconsistent performances, which may have been related. He all but admitted that he used Spider Tack on balls to gain an unfair advantage through an artificially enhanced grip, just days before the sticky substance was banned from the game. After that, whenever Cole pitched below his capabilities, observers wondered if it was because he did not have the adhesive stuff, and those doubts could continue well into the future.

Perhaps more significantly, Cole injured his left hamstring in a start against the Toronto Blue Jays on Sept. 7. His earned run average in his next four regular season starts was 6.35, and his troubles continued on Tuesday.

His Boston counterpart, Nathan Eovaldi, dominated the Yankees through five innings, but in the sixth Anthony Rizzo hit a towering home run that curled around the foul pole in right field. Then Aaron Judge reached base on an infield hit, which brought Alex Cora, Boston’s manager, out of the dugout.

Eovaldi had given up only two hard hits — a ball of the Green Monster by Giancarlo Stanton in the first and the Rizzo home run — and the Red Sox bullpen has been a spotty, at best, for most of the season. But the Boston relief corps had shown improvement in the final few weeks, and Cora called on Ryan Brasier, one of the team’s most reliable relievers down the stretch.

Brasier’s first assignment was to face Stanton, who hit another shot off the Green Monster. Phil Nevin, the Yankees’ third base coach, aggressively sent Judge homeward, but Enrique Hernandez fielded the ball cleanly and threw quickly to Bogaerts, whose relay to catcher Kevin Plawecki was perfectly on target for the second out.

Brasier got Joey Gallo to pop up for the final out. In the bottom of the inning, Boston regained its three-run lead on a Verdugo double that scored Bogaerts ahead of the tag.

Tuesday night marked the fifth time the Yankees and Red Sox had played an elimination game, in which the winner advanced and the loser’s season ended. The first was in 1949 when the Yankees won a regular-season tiebreaker. Then came 1978. In Game 7 of the 2003 A.L. Championship Series, Boone, then the Yankees’ third baseman, hit a game-winning homer in the 11th inning off Tim Wakefield.

But in 2004, the Red Sox drastically changed the script, coming back from an 0-3 deficit to win that series in seven games. Since then, the nature of the rivalry has shifted, and Boston has won four World Series titles, while the Yankees have claimed one, in 2009.