BOSTON — So this was how it ended for the Tampa Bay Rays, the team that pushes and pokes conventional thinking and usually gets away with it. The Rays set a franchise record with 100 victories this season, eight more than the Boston Red Sox. But when it mattered most, in their American League division series, the Red Sox upended the hoariest axiom of all.

Sometimes, good hitting really does stop good pitching.

A sacrifice fly by Kiké Hernandez vaulted the Red Sox into the American League Championship Series on Monday night, capping a 6-5 victory and a three-games-to-one series win. The Red Sox will face the Chicago White Sox or the Houston Astros in the next round, which starts on Friday, as they seek their fifth championship in the last 18 seasons.

“We always said we had a good baseball team that had some holes, and we still have some holes,” Manager Alex Cora said. “But at the end, for as bad as it looked sometimes, we’re still here. We’re still in the dance.”

It was a day for dancing in the Hub. In the morning, the locals lined the streets to toast the return of the Boston Marathon; at night, they packed old Fenway to celebrate a playoff clincher. It was hard to envision this just a few days ago, when the Rays’ sorcerers spun a shutout in the series opener. Who knew then that they’d cast their last spell?

The Red Sox hitters spent the next three days thrashing the Rays. They batted .341 for the series, striking out only 26 times. In the last three games, Tampa Bay starters never made it through the third inning — and even when the bullpen pitched well, Boston’s hitters were not fooled.

“We just could not create that swing-and-miss that we’ve done so well throughout the regular season,” Rays Manager Kevin Cash said. “They really had a good approach. It felt like there was constant pressure. There were no easy outs.”

The Red Sox scored the most runs in the majors during their championship seasons of 2004, 2013 and 2018, and their 2007 title team ranked fourth. This year they were fifth, and actually scored fewer runs than the Rays.

But the separator was strikeouts: the Rays’ hitters had the most of any winning team in the majors, while the Red Sox had among the fewest. Boston pitchers exploited the Tampa Bay hitters over and over again, generating 46 strikeouts and holding the Rays’ leading home run hitter, Brandon Lowe, hitless for the series.

The Rays had hoped to build off their run to the A.L. pennant last fall. But the Red Sox have a seasoned team, too, and Hernandez — who helped the Los Angeles Dodgers beat the Rays in the last World Series — said the Game 1 shutout jolted the lineup.

“Playoff baseball, if you don’t let the moment get too big, it kind of brings the best out of people,” said Hernandez, who hit .450 (9 for 20) in the series. “We were able to lock it in and make better decisions on the pitches we wanted to swing at, the zones we wanted to attack.”

Hernandez embodies his retooled team. For all of their success this century, the Red Sox have also finished last in the A.L. East four times. That includes last season, the first under Chaim Bloom, the team’s chief baseball officer and a former top executive with the Rays.

Bloom traded Mookie Betts but kept much of the well-paid core from the 2018 champions — Chris Sale, J.D. Martinez, Xander Bogaerts, Nathan Eovaldi. He has supported them with a series of low-cost investments, in the Tampa Bay style, and Hernandez is the most expensive import.

Hernandez, 30, signed for two years and $14 million after playing every position but catcher for the Dodgers. He had essentially been a Hollywood stuntman yearning for a leading role.

“But I didn’t want the easy way of playing every day,” Hernandez said. “I didn’t just want to play every day in the big leagues. I wanted to play every day where it mattered.”

In two trips to Fenway with the Dodgers, Hernandez said, he had never appreciated the atmosphere. At the icy 2018 World Series, the Dodgers lost twice and spent too much time shivering to look around. When Hernandez returned the next summer, the Red Sox were far out of first and the crowd seemed listless.

This season has been much different, especially now. The Red Sox are 3-0 at Fenway this postseason and have eliminated two division rivals, including the Yankees in the wild-card game. The Rays games were classics, with a leadoff and game-ending homer on Sunday — Kyle Schwarber to start, Christian Vazquez to finish — and a little bit of everything on Monday.

Rafael Devers ripped a three-run homer. The Red Sox led by five, then lost the lead. In the eighth, Cora called for Garrett Whitlock, a rookie with a rebuilt elbow who was plucked from the Yankees in the Rule 5 draft. Whitlock faced six hitters and retired them all.

He also got the win after a minimalist rally from a team that can also swing big: single, sacrifice bunt, infield single, sacrifice fly. Love that dirty water.

“I feel like nobody expected us to be here right now,” starter Eduardo Rodriguez said. “And look where we are.”

The Red Sox are onto the A.L.C.S. Opposing pitchers beware.