Lucic-Baroni, 34, won 6-4 6-2 to reach the last eight of a major for the first time since Wimbledon in 1999, when she lost to Steffi Graf in the semi-finals.
The Croat will next play fifth seed Karolina Pliskova, who beat 22nd seed Daria Gavrilova 6-3 6-3.
"I am a tough little cookie and really stubborn," said Lucic-Baroni.
"When I want something I will work hard and do anything I need to get it. What a satisfaction."
Lucic-Baroni did not play at the Grand Slams between 2003 and 2010 because of a series of personal issues, which led to her fleeing Croatia along with her mother Andelka and four siblings in the dead of night and heading to the United States.
"I had a rough patch in my life early on but I am really blessed with the family I have," she said.
She had been a tennis prodigy, winning junior titles at the Australian and US Opens, making her Grand Slam debut as a 15-year-old at the US Open and going on to win the Australian Open doubles title with Martina Hingis in 1998.
Financial problems resulting from the move to America meant she played just three matches between 2004 and 2006, but she returned to the sport and has finished in the world's top 100 for the past six seasons, reaching the last 16 at the US Open in 2016.
Now ranked 79th, her first-round win over Qiang Wang was her first Australian Open main-draw win since 1998.
If Lucic-Baroni wins her quarter-final, and Serena Williams beats Johanna Konta, the pair will meet for the first time since Wimbledon 1998.
"I always said I have the game. But to work so hard and make so many sacrifices, I hope no-one is going to pinch me and wake me up because this is just incredible," said Lucic-Baroni.
Mirjana Lucic-Baroni ended an 18-year wait for a Grand Slam quarter-final place by beating American qualifier Jennifer Brady at the Australian Open.