And there's no place they'd rather be to try to get it done, starting with Game 6, which is set to take place on Saturday at 8 p.m. ET on FOX.
"Yeah, they're real passionate about their sports in this town," said first baseman Mike Napoli. "I'm pretty sure they're going to be all fired up."
Napoli played in Fenway as a visitor in three postseasons during his years with the Angels. He's much happier to have all the noise at his back instead of in his face.
"Yeah, I mean, it's always fun playing here," he said. "I remember coming in here, it's intimidating. They get loud, they're on you. Just all the history here. It's definitely a fun place to play. Being on the good side now, it's sure nice to have them on your side."
The Red Sox had the best home record (53-28) in the AL this season. They hit better at home (.285 average, .819 OPS in the regular season and postseason combined) than on the road (.269, .773), and they also pitched markedly better at Fenway (3.54 ERA, .687 OPS against) than in enemy territory (3.99/.729).
"You know how good we play back in our building," said David Ortiz. "Hopefully, that's the case on Saturday."
Ortiz reminded one and all about the type of October magic that can happen at Fenway in the eighth inning of Game 2, when he seemed to reverse the course of this entire series by clocking a game-tying grand slam with two outs.
That isn't to say the Sox can simply throw on their home whites and that will be enough to beat the Tigers. It won't be easy. In fact, like everything in this series so far, it figures to be pretty difficult.
The Tigers present two pitchers any team would want to have lined up for Games 6 and 7 of an LCS -- Max Scherzer and Justin Verlander.
The Red Sox, however, also have two pretty good pitchers lined up in Clay Buchholz and John Lackey.
And perhaps even more important, they have Fenway, with the Green Monster hovering just 310 feet from home plate and all those various nooks and crannies throughout the outfield that can be challenging for defenders who don't make their home in Boston.
Why has this particular Red Sox team responded so well to Fenway?
"It's a mixture of the crowd, it's a mixture of a lot of guys who haven't played here before, they're getting that Boston atmosphere for the first time and are just kind of thriving on it," said catcher Jarrod Saltalamacchia. "Our front office did a great job of putting a group of guys together who thrive in these situations, where fans expect you to be great and they want you to be great."
The allure of playing at home is something that kept the Red Sox hungry in the days after they won the division title back in September. They knew that if they could get the best record in the AL, they would have home-field advantage in all three rounds of the postseason.
They accomplished that mission, and now they hope to reap the benefits.
"The one thing that we've looked at is, 'What's the task at hand, and what might that mean down the line?'" said manager John Farrell. "We didn't know at the time it would lead to a Game 6 of the ALCS, but it was important to us as we were finishing out the final week of the regular season to secure home-field advantage."
But it isn't just the comfort of home that inspires the Red Sox. There is also the crowd, and a ravenous one it will be. Boston fans might have lost a tiny bit of their collective edge in 2008 and 2009, thanks to the championships of 2004 and 2007.
But as Rays right fielder Wil Myers can attest from a couple of weeks ago, Red Sox Nation is blood-thirsty these days.
And here's an interesting tidbit heading into Saturday's game. The Red Sox are 5-0 all-time in Game 6 at Fenway. The most memorable of those five triumphs was the epic contest against the Reds in 1975, when Bernie Carbo hit a pinch-hit three-run homer in the eighth, and Carlton Fisk hit the foul pole in the 12th.
The last Game 6 at Fenway was in the 2007 ALCS, when J.D. Drew belted a grand slam and Curt Schilling won yet another big game, forcing Game 7 against the Indians.
This edition of the Red Sox will try to add a Game 6 memory.
"You know the fans here, man. They go ballistic," said Xander Bogaerts. "I think we get the kind of atmosphere where we feed off these fans."