Zheng clinched her place in the second week at Melbourne Park with a 6-4, 2-6, 7-6 win over compatriot Wang Yafan on the same Rod Laver Arena court where Li lost the 2011 and 2013 finals before taking the crown in 2014.
"I was super happy to meet her and have the chance to talk with her because I never talked with her in person. That was really special for me," Zheng told reporters.
"I think she's much more beautiful she looks on TV. She said, 'don't think too much, just keep it simple'. I think that's what I need to do right now."
Zheng, who moved to Li's hometown of Wuhan to train at the age of eight, said she watched her hero beat Dominika Cibulkova in the 2014 final at least 10 times.
Li's triumph was the first by a Chinese or Asian player at Melbourne Park and Zheng's chances of emulating her, although still modest, have been greatly enhanced by a culling of seeds in her quarter of the draw.
The 12th seed will next meet world number 95 Oceane Dodin with either Italian 26th seed Jasmine Paolini or world number 75 Anna Kalinskaya awaiting the winner of that tie in the quarter-finals.
When China was investing heavily in tennis as part of a government drive to develop a multi-billion dollar sports industry, the Australian Open started marketing itself as the "Grand Slam of the Asia-Pacific".
It remains the closest thing to a home Grand Slam for Chinese players and Zheng said she had felt the support of her compatriots on court this week.
"The Australian Open has a lot more Chinese fans, so I feel like I'm playing at home because the crowd was supporting me more," she said.
"In the US Open, it's more like half and half. Sometimes they support opponents more, sometimes it's me. It's a totally different atmosphere. It's the first time I've reached the fourth round here, so that was special."
