Venus Williams will face a player she inspired to take up tennis after reaching her first Grand Slam quarter-final since 2010 in the Australian Open.

Williams defeated sixth seed Agnieszka Radwanska 6-3 2-6 6-1 on Rod Laver Arena to set up a last-eight clash with American teenager Madison Keys, who had earlier beaten compatriot Madison Brengle in straight sets.

Keys, 19, has made no secret of her admiration for the Williams sisters, although it is something the siblings might not like being reminded of too often.

“Apparently she started playing because she watched Serena and I,” said the 34-year-old Venus. “She was watching me in diapers.”

Venus’s last quarter-final appearance in a Grand Slam came in the US Open in 2010, the year before she was diagnosed as suffering from Sjogren’s syndrome, a debilitating autoimmune disease.

Keys was just five years old when Venus won the first of her seven Grand Slam singles titles at Wimbledon in 2000 and is relishing the prospect of facing one of her childhood heroes.

“I’m just really excited,” she said. “I think it’s a huge opportunity for me. I haven’t been in this situation before. I’m going to make the most of it. I want to be at the end of the tournament holding the trophy up. That’s my goal in the long run.

“I think Venus has helped the sport, especially the women’s side with equal prize money. She was a huge part of that. Just watching her is inspirational. She’s had her health battles but she loves tennis. She’s still out here and she’s doing it remarkably well. I hope I can be someone similar to that.”

Keys could have to face both Williams sisters if she wants to win the title, with Serena taking on Dominika Cibulkova in the other quarter-final in the top half of the draw.

Top seed Serena had some help from her fans to thank for overcoming another slow start – and a coughing fit – to defeat Spain’s Garbine Muguruza in three sets and gain a measure of revenge for her defeat at the hands of the 24th seed in the second round of last year’s French Open.

Muguruza won 6-2 6-2 at Roland Garros and threatened to do the same here as she won four games in a row from 2-2 in the opening set.

Williams, who had to save three set points against Vera Zvonareva in round two and then came from a set down to defeat Elina Svitolina, won the second set 6-3 to level the match and then crucially held serve at the start of the decider.

The world number one had to save six break points in an epic game lasting 13 minutes and promptly broke a disheartened opponent on her way to taking the set by a flattering 6-2 scoreline.

Williams admitted the shouted advice of one spectator to “use spin” helped her turn the match around, adding: “She (Muguruza) played so well and she did everything she needed to do in the first set.

“I just decided to do my best and stay focused and stay relaxed. She hits the ball really, really big and hard, but someone in the crowd was like ‘Use spin’. I was like ‘Okay’ and it really helped me.”

Williams struggled with a cough during the match and the 33-year-old said she had been “a little sick the past couple of days”. She added: “I’ve been fighting it with vitamin C and all kinds of stuff. I definitely feel better than yesterday.”

Cibulkova reached the last eight after a see-saw battle with former champion Victoria Azarenka under the roof on Rod Laver Arena.

A morning rain shower quickly cleared but the roof remained closed for the duration of a hard-fought fourth-round clash which Cibulkova eventually won 6-2 3-6 6-3.

“I just walked on court and all the great memories come to my mind,” said Cibulkova, who lost to Li Na in last year’s final. “I just have to believe in myself and that’s what I am doing right now.

“The first set I was killing her from the forehand and then she started stepping inside the court and to serve much better. I felt like I had to put in something extra and play really, really good and that’s what I did.”

Azarenka, who won the title in 2012 and 2013 but was unseeded this year due to an injury-plagued 2014, led 2-0 in each set but lost six games in a row in the opening set and six out of seven in the decider.

“I take it as progress,” she said. “I think there are a lot of positive things to take from here. It’s a good start. There was some good quality tennis, even though the result is not the one that I wanted.

“But I need to be realistic a little bit and keep working hard and try to sharpen my game. I need to be more consistent and I need to be able to take my opportunities when I have them. Overall I can be pretty happy but I’m such a perfectionist that I don’t want to be satisfied.”

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