MELBOURNE, Australia — As a Pittsburgh Steelers fan whose parents ended up buying the N.F.L.’s Buffalo Bills, Jessica Pegula has had to adapt. But she is in deep now, extolling the leadership virtues of quarterback Josh Allen even as she competes in the Australian Open tennis tournament, and taking the court in an outfit whose red, white and blue hues summon the Bills’ colors, thanks to her sponsor thinking ahead.

“It was so random, but I’m like this is perfect,” Pegula said.

She even signed the camera lens after her third-round singles victory with a tidy note that read: “Bills you’re next.”

“I’m like come on, I backed myself up, now you guys got to get the win,” Pegula said with a chuckle ahead of the Bills’ divisional playoff game against the Kansas City Chiefs on Sunday.

Stacking up wins would be an outcome to savor for the Pegula family, and Jessica has provided another strong run down under.

It was in Australia that she launched her breakthrough season in 2021 by reaching the quarterfinals. At 27, she is on the verge of leaping back into the top 20 whether she wins or loses in her fourth round rematch with Maria Sakkari, who saved six match points before beating Pegula in the round of 16 at the Miami Open last year in a memorable topsy-turvy duel.

But Pegula has bounced back from worse. A child of privilege by her own admission, she has shown perseverance and pluck in her quest to become a Grand Slam contender. Yes, she had access to private coaching and abundant support from her family: her 70-year-old father Terry is a billionaire businessman who made his $5.7 billion fortune primarily in natural gas and in real-estate development.

But Pegula had to overcome major knee and hip surgery in her late teens and early 20s that required extensive rehabilitation before she finally broke into the elite.

“She was on her way up twice and had to start over again,” said Michael Joyce, who coached her for six years, beginning in 2011 after coaching Maria Sharapova. “Jessie could easily have thrown in the towel obviously with her family and her situation, and the fact that she kept coming back was special. A lot of people would have said, ‘Screw this, I’m done,’ especially in her position.”

Tennis, with significant coaching and travel costs, is an expensive sport to master at a high level, but highly ranked stars from ultra rich backgrounds are rare on the tour. Pegula is perhaps the first on the women’s tour since Carling Bassett, daughter of Canadian brewery executive John Bassett, broke into the top 10 in the 1980s.

“I know a lot of people from very wealthy families who are pretty good, good enough to play in college or something, but they usually fizzle out,” Joyce said.

Pegula said she has sometimes felt self-conscious about her family’s wealth, concerned it might make others uncomfortable. Joyce said she was often hesitant to organize training sessions with outsiders at the family’s luxurious home in Boca Raton, Fla., with its two tennis courts — clay and hardcourt.

“I was maybe kind of trying to hide it a little bit,” Pegula said. “Then I think I kind of embraced it a little bit, not like over the top, but I think once I became more comfortable and I knew I was doing the hard work and all that I was, like, hey I do have a different story but maybe it’s kind of a cool story. Maybe it’s OK if I embrace the Bills and the teams a little bit more and stuff like that.”

She added: “But I’ve always been kind of low key. I don’t like to flaunt, and I think that’s why I’ve been able to be successful, too.”

Terry and his wife, Kim Pegula, who was born in Seoul and grew up in Fairport, N.Y. near Rochester, bought the N.H.L.’s Buffalo Sabres in 2011 when Jessica was turning 17. They purchased the Bills in 2014 for $1.4 billion.

It was not until then that Pegula said she became acutely aware of her family's fortune, but it did not change how she felt about tennis.

“I’ve always been super driven, before the Bills and the money and all that stuff,” she said.

“This is always what I wanted. So, when all this stuff happened to me later on in my life, people would ask me, ‘Why are you doing this?’ And I’d be like, ‘I don’t understand. This hasn’t changed since I was 6 or 7 years old. Why would it change now?’”

Pegula said she has come to believe that she has a responsibility to do justice to her advantages.

“I’m given this amazing opportunity. Why would I want to sabotage that if I really love what I do?” she said. “I don’t shy away from the fact that people don’t get as many opportunities, and I think people are more realizing that giving everyone equal opportunities is important. But I didn’t choose the life I was supposed to have. You are kind of born into it, and I think everyone is dealt a different hand. It’s how you deal with it, and I’m glad that I was able to do it justice and not take it for granted. To me, it would be selfish to do a disservice to that.”

Pegula said she has learned to “embrace the grind” — the fitness training, practice sessions and preventive work now required to keep her healthy after the injuries that could have ended her career.

At 5-foot-7, she is not the most imposing athlete on a women’s tour increasingly inhabited by taller players with explosive power and movement. But she has exquisite timing, excellent fundamentals, a fine grasp of tactics and an even temperament.

“It used to drive me nuts,” Joyce said. “She could go through a whole tournament without one fist pump.”

Equanimity can be useful in a brutally competitive sport where success is precarious. One of Pegula’s closest friends, Jennifer Brady, was an Australian Open finalist last year but has now missed the last two majors with a chronic foot condition.

It can all seem fragile, all the more so given the coronavirus pandemic. Pegula married her longtime boyfriend, Taylor Gahagen, in October at the Biltmore Estate in Asheville, N.C., but her coach, David Witt, tested positive for the coronavirus and she, as a close contact, withdrew from the Billie Jean King Cup team event.

The next day she tested positive. So did her husband. “We had a Covid honeymoon basically,” Pegula said. “We were in our house for two weeks.”

Though Pegula said it took her “a few weeks” to recover, she enjoyed the extended off-season and the chance to spend time with her three dogs in Boca Raton: Maddie, a miniature Australian shepherd; Dexter, a German shepherd; and Tucker, a chocolate Labrador.

“A lot of different personalities,” Pegula said. “Like three kids I guess. But you have to adapt.”

Consider that her catchphrase. In earlier days, she had a dog named for Sidney Crosby, the Pittsburgh Penguins hockey star. The Pittsburgh connection was real: Her father is from Pennsylvania and graduated from Penn State. Though Jessica was born in Buffalo, the Pegulas lived in Pittsburgh when she was young.

“We were really not Bills fans to be honest, but that’s obviously flipped,” she said, preparing to check the time difference carefully from Australia and watch Sunday’s big game.