MELBOURNE, Australia — Serena Williams saw that older sister Venus was progressing to the fourth round at a Grand Slam tournament for the first time since 2011 and it inspired her own comeback win on Saturday at the Australian Open.

Top-ranked Serena Willliams made a lackluster start to her third-round match against No. 26-ranked Elina Svitolina on Rod Laver Arena – knowing when she walked out that Venus was down a set and a break on a nearby court – but picked up her game after realizing her sister had recovered and was advancing with a 4-6, 7-6 (3), 6-1 win over Camila Giorgi.

Svitolina “kept hitting winners in the first set, there’s not much I can do. Then I saw (Venus’) score and thought ‘Wow she’s winning, well I can do better,”‘ Serena Williams said after her 4-6, 6-2, 6-0 win.

“We always motivate each other.”

She’ll have to be at the top of her game in the next round she meets No. 24 Garbine Muguruza, who beat her in the second round at the French Open last year. Muguruza defeated Timea Bacsinszky 6-3, 4-6, 6-0.

Venus Williams couldn’t contain her delight after her win on Margaret Court Arena, beaming a smile as she turned to wave to all sides of the stadium and then moved to the music blaring over the loud speakers.

The 34-year-old, seven-time major winner next faces sixth-seeded Agnieszka Radwanska, a semifinalist here last year and Wimbledon finalist in 2012, who advanced with a 6-0, 7-5 win over Varvara Lepchenko.

Venus Williams was diagnosed with an auto-immune condition called Sjogren’s syndrome in 2011, which can cause joint pain and saps energy. In the 13 Grand Slam tournaments after her fourth-round appearance at Wimbledon that year, she skipped two and made first-round exits in four others, including the 2014 Australian Open.

She was only two points from a third-round departure in the second set against Giorgi, serving at 30-30 in the ninth game, before working her way back into the contest. Giorgi contributed to her own demise with 16 double-faults.

“Well, this old cat has a few tricks left,” Williams said in her on-court interview. Being back into the fourth round, “feels fantastic especially when things happen in your life and it’s not in your control, so it definitely feels awesome to be here, and I don’t want to leave it at that. I’ll try to keep going.”

Late Friday night, Roger Federer was eliminated in the third round by Italian Andreas Seppi 6-4, 7-6 (5), 4-6, 7-6 (5).

Federer was trying to come from two sets down for the 10th time in his career, most recently at last year’s U.S. Open against Gael Monfils.

The 17-time Grand Slam champion hadn’t been beaten this early at the Australian Open since 2000, the second year he played here. It was also the unseeded Seppi’s first win against Federer in 11 matches.