ST. PETERSBURG, Fla. — The Tampa Bay Rays had just won again Tuesday night, but their architect was flustered by a small crack in the foundation. One of the Rays’ pitchers had taken a line drive off his foot, another had a forearm problem, a few could have used a break, and nobody from the minors quite fit as a replacement.

Erik Neander, the Rays’ president of baseball operations, wondered if any available veterans were lined up to start in the minors the next day. He remembered an email he had recently received from the Cincinnati Reds: Chase Anderson, a well-traveled, 35-year-old right-hander for Class AAA Louisville, was available.

Anderson had pitched for a month in the Rays’ farm system last summer — and, most important, was lined up to start a morning game in Omaha in about 12 hours. Perfect. The Rays made the requisite phone calls, got Anderson for cash considerations — “Maybe a dollar,” Neander said — and booked him on a 7 a.m. flight on Wednesday.

That night, naturally, Anderson pitched three scoreless innings against Pittsburgh for his first career save in the majors. Another victory on Thursday gave the Rays a three-game sweep of the Pirates and a spellbinding 26-6 record — the best 32-game start in the league since the 1984 Detroit Tigers, who won the World Series.

Even those Tigers, though, did not pummel their opponents this thoroughly, this early. The Rays, who will host the Yankees at Tropicana Field this weekend, had outscored opponents by 113 runs before Thursday’s 3-2 win over Pittsburgh. The last team with a wider run differential through 31 games: Honus Wagner’s Pirates in 1902, a year before the creation of the World Series.

“Everybody has confidence, everybody knows what we’ve got to do,” said Harold Ramírez, who floated through four organizations before becoming a .300 hitter for the Rays. “It’s like, a lot of people think we are a weird team because we do weird things. But I know everything we do is right.”

Everything? It’s not a stretch. The Rays have scored the most runs in the majors, with the highest batting average, the best on-base plus slugging percentage and the most home runs. Even while using the most pitchers in the majors, the Rays have allowed the fewest runs and homers and the lowest opponents’ average and O.P.S.

“They just kind of create these pitchers out of nowhere, like people haven’t heard of them before,” said the right-handed starter Zach Eflin, who pitched in the World Series for Philadelphia last fall and eagerly signed with the Rays in free agency. “There’s been a long kind of rumor in the big leagues, like there’s something in the water in Tampa — they know what they’re doing down there.”

The three-year, $40 million contract for Eflin, who is 4-0 after his victory on Thursday, was the richest free-agent deal in Tampa Bay history. Yet for all of their bargain shopping and roster churning, the Rays now stand out for something traditional and reassuring: Stability, in both leaders and players.

Kevin Cash became the team’s manager in December 2014, when only one of his peers (Cleveland’s Terry Francona) was in his current position. The hitting coach Chad Mottola and the pitching coach Kyle Snyder are in their sixth season together; no other team has had the same people in those roles for that long.

Then there’s the roster, with 17 players (including those on the injured list) who were part of the organization in 2020, when the Rays won the American League pennant.

“When you get a group that’s able to play together for a while, it’s kind of how the March Madness teams are,” the right-handed reliever Pete Fairbanks said. “When you get into the tournament and you’ve got a team full of juniors and seniors, those teams typically seem to have the ability to make noise.”

The Rays are more than a collection of role players who know the system. The left-hander Shane McClanahan, who is 6-0, started the All-Star Game last summer at Dodger Stadium. While the Yankees have no everyday players with a .900 O.P.S., the Rays have six, including shortstop Wander Franco, a switch-hitting force, and the stylish left fielder Randy Arozarena, who hit his eighth homer on Thursday.

Franco, 22, is signed through 2032, and while Arozarena, 28, does not have a long-term deal, he cannot be a free agent until after the 2026 season. His payday might come elsewhere.

“I know I play for a small-market team, but obviously I play hard and I do the things I do to be able to get that contract — and eventually I think it will happen, regardless of if it’s here or not,” Arozarena said in Spanish through an interpreter. “I know we’re in the business of the big leagues, but regardless of the market, all teams have a lot of money.”

The Rays, as usual, are getting a lot for their money: Their player payroll ranks among the majors’ lowest — roughly $73 million on opening day, ahead of only Baltimore and Oakland — but they have dominated baseball in nearly every category.

The hitters lead the A.L. in barrel percentage, hard-hit percentage and exit velocity, and the pitching staff is the only one in the majors to rank in the top three in all those categories. Statcast confirms those figures, but it’s a time-tested formula: Hit the ball hard and keep your opponents from doing the same.

“It looks a lot more complicated from the outsider’s perspective than it actually maybe is inside,” Snyder said.

The Rays’ schedule is getting much tougher now. Their opponents in March and April had a combined .427 winning percentage through Wednesday, but none of their May opponents had a losing record entering the month. But the Rays — who have made the playoffs four years in a row — are finding that nobody doubts them anymore.

“Before, it was: ‘We’re this, we’re that, we’re a gimmick,’ and suddenly, this year, we’re good,” Mottola said, speaking of the national impression of the Rays. “And it’s almost a little concerning — you’re going, ‘Wait, wait, we have to keep that underdog mentality.’ If we get stale and complacent and believe that we’re good, then we lose what we’re good at — if that makes sense.”

The Rays are good at being different. The owner Stuart Sternberg, a former partner at Goldman Sachs, cultivated a Wall Street-savvy front office in the mid-2000s, and Neander worked for Sports Info Solutions, an analytics hub co-founded by John Dewan and Bill James, before joining the Rays in 2007.

But while it’s tempting to credit data for the Rays’ success, it also misses the point. A team can embrace information while also respecting the pulse of the players. For the Rays, that is actually the secret to everything.

The night they scrambled to find Anderson, the Rays used seven pitchers. Only one faced the same hitter twice. The Rays are famous for this strategy; the opener, as it is known, is probably their best-known innovation. Yet the data behind it is mostly a front.

“It’s about putting players in position to have success and complete their competitive day feeling good about themselves and wanting to contribute more,” Neander said. “You’re trying to foster confidence. The ideal space is when everyone feels like they can do more.”

The baseball season is very long, of course. But few teams, to this point, have ever done more.