Dayana Yastremska can't convert on the easy shot as Serena Williams dominates her way to a third round 6-2, 6-1 win. (0:27)

MELBOURNE, Australia -- It was all a bit overwhelming for the latest opponent who could do nothing to slow Serena Williams at the Australian Open. So Dayana Yastremska, an 18-year-old from Ukraine, found herself wiping away tears as she walked to the net.

Williams knows what it's like to be the one weeping after a loss, so she put her right hand on Yastremska's shoulder and consoled her by saying, "You're so young. You did amazing. Don't cry."

Then they embraced, and Williams patted Yastremska on the back. Maybe Williams will be tested in the fourth round, because no one has come close to making her work too hard so far, including for the 6-2, 6-1 victory on Saturday.

"I could tell she was quite upset. I kind of liked that. It shows she wasn't just there to play a good match -- she was there to win. She wanted to win. That really broke my heart," Williams said. "I think she's a good talent. It's good to see that attitude."

Williams told Yastremska as much in the locker room afterward.

"She said, like, 'You're young, you're very good and you will be a good player in the future. It's nice to hear those words from a legend," said the 57th-ranked Yastremska, who eliminated 2011 US Open champion Sam Stosur in the first round and 23rd-seeded Carla Suarez Navarro in the second.

"If she thinks so," Yastremska added about Williams, "then maybe that's true."

Serena Williams hits a return against Dayana Yastremska at the Australian Open on Saturday. Jewel Samad/Getty Images

Now, Simona Halep is next. The No. 1-seeded Halep, last year's runner-up at Melbourne Park, reeled off six consecutive games in one stretch and ended up beating Venus Williams 6-2, 6-3 in the third round.

On facing Serena, with a quarterfinal berth at stake, Halep said after defeating Venus, "It's going to be a bigger challenge. I am ready to face it."

After two tough three-set tussles in Australia, Halep had a much easier time of things Saturday and made only 12 unforced errors, while Venus made 33. Halep played with her left thigh taped but moved around the court well.

For Venus, this is the fifth consecutive Grand Slam tournament she exits before the fourth round.

Versus Yastremska, Serena Williams grabbed a pair of service breaks and a 4-0 lead after less than 15 minutes and was well on her way to yet another easy-looking win.

Not only has she won every set she played this week -- and 20 in a row at Melbourne Park, dating to the start of her 2017 run to the title (she sat out last year's tournament after having a baby) -- but Williams has ceded a total of only nine games through three victories.

The woman Serena Williams lost to in last year's chaotic US Open final, Naomi Osaka, dusted herself off after tumbling to the court and erased a big deficit by taking 11 of 12 games to beat No. 28 Hsieh Su-wei 5-7, 6-4, 6-1.

No. 6 Elina Svitolina also put together quite a comeback to win 4-6, 6-4, 7-5 against Zhang Shuai, while No. 13 Anastasija Sevastova eliminated No. 21 Wang Qiang 6-3, 6-3.

"I was ... struggling," said Svitolina, who was treated by a trainer after the second set. "I just told myself, 'I'm going to die or win.'"

Svitolina will face Madison Keys, who reached the second week for the fourth time with a 6-3, 6-2 win over No. 12-seeded Elise Mertens.

Keys, a semifinalist at the Aussie Open in 2015 and runner-up at the 2017 US Open, converted four of her seven break-point chances and saved one of the eight she faced against Mertens.

"I'm definitely very happy with how I played today -- especially how I served and got out of some could-be-bad situations," she said.

The No. 17 seed didn't play any warm-up tournaments before the season's first major and says, "I'm feeling a lot better now that I've had three wins."

She has reached the semifinals at three of the past five Grand Slam events.

Karolina Pliskova, the 2016 US Open runner-up and No. 7 seed eliminated 27th-seeded Camila Giorgi 6-4, 3-6, 6-2.

Two-time major champion Garbine Muguruza moved on to the fourth round, defeating Timea Bacsinszky 7-6 (5), 6-2.

Muguruza improves to 12-1 in the third round of majors (.923), her highest win percentage of any round.

Pliskova next takes on Muguruza for a quarterfinal berth. Pliskova holds the edge in their career head-to-head matches at 7-2, but Muguruza won their only previous match at a major (2013 French Open).

Information from ESPN's Stats & Information and The Associated Press was used in this report.