The 34-year-old Croatian defeated her third-seeded Polish rival 6-3, 6-2 in 63 minutes, reports Efe.
Radwanska is the second favourite female player to be eliminated from the tournament after Romanian Simona Halep, who fell in the first round against Shelby Rogers of the US.
"It's amazing, it's been so long since I won a match or two... actually this is the first time ever in Australia. It's an incredible feeling," Lucic-Baroni was quoted as saying by tournament's official website.
"I knew what I had to do. It's just a matter of getting on court and doing it and I was able to execute perfectly tonight. I'm super thrilled," the Croat explained.
Lucic-Baroni will now play the winner of the match between France's Alize Cornet and Greece's Maria Sakkari in the third round on Saturday.