And with a typical late burst, they did. Trailing by two runs heading into the eighth, the Yankees put the tying run at third base when Alex Rodriguez and Hideki Matsui singled, and Nick Swisher reached on first baseman Chris Richard's throwing error.
After Robinson Cano struck out, Posada launched a Grant Balfour pitch over the right-field wall, giving the Yankees their first lead and helping Jeter appreciate his accomplishment.
"Obviously, when you're in this type of situation and you're talking about personal things, it's a lot harder to enjoy it if you don't win the game. When Jorge hit the big home run, I think it makes everyone a little bit happier on the team."
Jeter, though, did not yet feel secure -- and for good reason. The Yankees, exercising caution, opted to proceed without Mariano Rivera, who has been battling groin stiffness but otherwise would have closed out the game.
Instead, Brian Bruney came on to begin the ninth, walking the first batter and serving up a deep flyout to straightaway center field to cleanup hitter Ben Zobrist. After another flyout, manager Joe Girardi called on lefty Phil Coke, who struck out the left-handed-hitting Richard to end the game.
In addition to solidifying the Yankees' lead for home-field advantage in the playoffs, the four-game sweep all but knocked the Rays out of postseason contention.
Any talk of playoff baseball in the Bronx, of course, is not complete without a discussion of Joba Chamberlain, Wednesday's starting pitcher. Shackled by a strict innings limit, Chamberlain threw just three innings and 55 pitches, allowing two runs in the first but retiring the final eight batters he faced.
If the Yankees do secure home-field advantage throughout the playoffs -- and with three weeks to go, they hold a sizable lead -- they will be able to choose between a seven-day or an eight-day American League Division Series schedule. In effect, the Yankees may be able to choose between using three starters or four in the first round, a decision that could temporarily send Chamberlain back to the bullpen.
When asked about that scenario recently, Girardi remained mum. All Chamberlain can do is pitch as well as possible now, in his first full season as a starter -- but that is difficult to do with the Yankees intent on limiting his innings.
"I was getting a little frustrated because I was throwing well," Chamberlain said, "but it was definitely my best start overall in a while, so I was pretty happy with it and happy to be part of this game, and more importantly the win."
That came even despite the fact that Jeff Niemann, the Rays' starter, turned in the best performance of any pitcher in the game. Allowing three hits to Jeter and five to everyone else, Niemann struck out eight and walked just one. But he allowed the first hit of the winning rally before the Rays' bullpen and defense undermined his sharp performance.
Niemann pitched with a lead all evening thanks to Jason Bartlett's leadoff homer in the first and Pat Burrell's RBI single. But after Alfredo Aceves fired three scoreless innings of relief to keep the Yankees in the game, it was Bruney, not Niemann, who pitched with the lead in the ninth -- and Jeter who appreciated it most.
Posada's home run won the game, the team's 12th victory in 14 games, putting a positive stamp on what was already a remarkably positive night. It also gave the Yankees the most touching moment to date at their new stadium, and the hope that October -- once they arrive there -- will provide many more of those moments.
"I think now, the most important thing for everyone on our team is to remember first, we haven't accomplished anything yet," said Jeter, the man who on Wednesday accomplished a heck of a lot.