Lucic-Baroni also beat Radwanska at the Australian Open in January, in the second round en route to the semi-finals, where she eventually fell to Serena Williams.
"She was playing unbelievable tennis today," Radwanska, speaking about Lucic-Baroni, told the WTA.
"She was very solid from the beginning, very aggressive, hitting everything with unbelievable speed. I couldn't do much.
"The first set went so quickly, and in the second set I had chances, but I didn't come back and unfortunately I missed the chance to stay in the match. She was just too good today."
Another high seed had a fright, number six Garbine Muguruza overcoming a slow start to beat China's Zhang Shuai 4-6 6-2-6-2 in over two hours.
"Since I've started the year, I've had a lot of matches like this, and I'm expecting that every time I go on court," Muguruza said.
"A win is a win, I don't wish to have matches this tough every day, but I'm very happy with the way I'm facing them.”
It was Muguruza's second victory over Zhang in barely a month, after their match in Doha in February that also went three sets.
Second seed Karolina Pliskova's match went according to form, the Czech beating Kazak 27th seed Yulia Putintseva 7-5 6-3.
But 17th seed Anastasia Pavlyuchenkova was sent packing, beaten 4-6 6-0 6-3 by American Bethanie Mattek-Sands.
(Reporting by Andrew Both in Cary, North Carolina; editing by Sudipto Ganguly)
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