MELBOURNE—Nineteen-year-old Eugenie Bouchard stood at the baseline and squinted up at the bright blue sky above Rod Laver Arena. Her quarterfinal with Ana Ivanovic had rocked steadily back and forth for the last two hours, the momentum flowing from one side of the net to the other like water on the floor of a leaky boat. Their play wasn’t brilliant, but neither player had gone down with the ship, either. Ivanovic won the first set 7-5; Bouchard returned the favor in the second set by the same score. Now the deciding set had reached its crux.

For the first six games of the third, Bouchard, the 30th seed, had outplayed the former No. 1 and was now serving at 4-2. By the laws of this match, as well as many thousands of other tennis matches over the centuries, it felt like it was time for the momentum to shift again, from Bouchard to Ivanovic—from the hunted to the hunter. But Bouchard defied the sport’s psychological laws by winning the first three points of her service game to go up 40-0. Watching from the press section with my friend Tom Tebbutt, a Canadian tennis writer, I thought for a second about boldly declaring, “This is over,” but held my tongue. That’s usually the prudent move in high-anxiety second-week Grand Slam matches like this, and Bouchard proved me right by losing the next two points to make it 40-30. Instead, it was Tom who leaned over to me and said, “The fat lady isn’t ready to sing just yet.” In other words, this wasn’t over.

Bouchard tossed the ball and took a confident swing at her serve; it landed deep in Ivanovic’s backhand corner but not too close to either line—right where she wanted it. When Ana’s return floated back, Bouchard moved in and sent a forehand to the opposite corner. On her next shot, she ran Ivanovic back across the baseline with a backhand. Finally, Bouchard stepped forward to finish the rally with an old-fashioned punch forehand volley that Ana flailed at in vain. None of Bouchard's shots had been especially risky or spectacular, but she had controlled the rally throughout. More important, she had held serve and, finally breaking the match-long pattern, kept the momentum to herself. A few minutes later, she rifled her 47th winner down the line—again, not too close to the line—for a 5-7, 7-5, 6-2 win.

There in a nutshell is why Genie Bouchard is into the semifinals in her first appearance at the Australian Open, and just her fourth major overall. She plays within herself. She doesn’t get too up or down. She doesn’t let her emotions get the best of her. She doesn’t look toward her players' box much. She builds points rather than going for quick winners. She hits with compact, uncomplicated strokes and takes the ball early, at the top of the bounce. And—something that will make her popular with a lot of people—she doesn’t grunt. None of these eternal tennis virtues are ones you would expect to find in a 19-year-old; and all of them are difficult, if not impossible, to teach.

Never fear, though, Bouchard is not a 35-year-old disguised as a teenager. Australian fans found that out when she was asked, after the match, who she would date if she had her choice of anyone in the world.

“Justin Bieber,” Bouchard said, a little sheepishly, to a storm of boos.

By the time she reached a packed media-interview room an hour later, Bouchard was back to business. With her deep voice, near-total lack of giggle, and seeming disdain for blabber, she's as serious and controlled in press conferences as she is on the court. And from what she says, she’s as unruffled on the inside as she appears to be on the outside.

“I tried to stay calm,” Bouchard said when she asked about the pressure of trying to reach a Slam semi. “I tried for sure to show I was calm. I did feel confident. Having lost the first set, I just focused on what I had to do during the point to try to win, just try to keep pressing her and moving forward. That’s what kept me really calm.”

Asked if there was a “pinch me, I'm dreaming” aspect of this win for her, Bouchard said no. The semis of a Slam at 19 is no big stunner for her.

“It’s something I’ve been doing since I was five years old," she said, "working my whole life for and sacrificing a lot of things for. So it’s not exactly a surprise. I always expect myself to do well. I’m happy just to have gone through this step. I’m not done.”

Someday, Bouchard may recall those words and chalk them up to the blind confidence of youth—“I didn’t know how hard it was to do what I did,” I can hear her saying when she looks back on this moment in 10 years. For today, though, Bouchard didn’t betray any doubts, or even much emotion. She concluded one answer by stating, “I have a match on Thursday.” She sounded about as excited as if she were trying to reschedule an appointment with her dentist.

A native of Montreal, Bouchard’s father is an investor. Her family lives on the same street as former Canadian prime minister Brian Mulroney, and has an obsession with English royalty—obscure British royalty. Genie and her fraternal twin sister, Beatrice, are named after Prince Andrew’s daughters. Bouchard’s upper-class, Anglophile background may explain her poised, somewhat aloof, Grace Kelly-with-a-Babolat demeanor. Though it wouldn’t explain the Gangnam-style video she made with friend Laura Robson last fall. Genie is coached in Florida by former pro, and noted stickler for technique, Nick Saviano, but she’s not a total grind.

There will be plenty of time—years, hopefully—to analyze her game, so I’ll just point out one shot that I found striking today, her backhand. This is a remarkably compact yet versatile stroke, the near-equivalent of a second forehand. Sometimes Bouchard looks like a left-handed baseball player swinging for the fences; other times she looks like a cricket batswoman fending off a tricky ball; and when a return of serve comes straight at her, she can look like a hockey goalie slapping the ball out of the crease. On one running passing shot, Bouchard shortened her swing and bunted the ball deftly past Ivanovic. On others, she leaned in and cracked it short and cross-court, a tough angle to create. And she frustrated Ivanovic by blocking her best serves and ground strokes back into the open court. As with all two-handed backhands, Bouchard’s is deceptive.

“She’s a young girl,” a suitably impressed Ivanovic said afterward. “I think she has a very bright future in front of her. She’s a very aggressive player. It’s sometimes hard to read her game. There’s really no patterns like with other players. She’s a great mover.”

Last year, a 19-year-old American, Sloane Stephens, announced herself by reaching the semifinals here. This year Bouchard has done the same thing at the same age, to even greater fanfare. (She's been loudly supported at every match by "Genie's Army," a group of—mostly young male—Melburnians that she had never met before the tournament.) Bouchard and Stephens have opposite strengths and weaknesses. Sloane has putaway power and pop on her serve, but lets her emotions get the best of her; Eugenie gets few free points, but always remains composed. There’s no reason to think that she won’t have similar ups-and-downs to Sloane’s; just last week, Bouchard lost to Bethanie Mattek-Sands in straight sets in Sydney. And she’ll have her work cut out for her against two-time Aussie finalist Li Na in the semis.

But Bouchard, known for her looks, is a lovely player to watch not because of the natural beauty of her shots, but because of the canny way she combines them to win points. She's one of the rare young players whose game is more than the sum of its parts.