Lyubtsova qualified for her first main draw on the WTA Tour, then won only four points on her serve.
In Radwanska's first appearance in this tournament, she also lost as a qualifier in the first round in 2007. Then she went on to win four titles - three last year - and become the first Polish woman to make the top 10.
Fourth-seeded Radwanska will next face Alisa Kleybanova, a 19-year-old Russian who reached the fourth round at the Australian Open. Kleybanova defeated Tsvetana Pironkova of Bulgaria 6-2, 7-6 (3).
The only other seed in action, No. 5 Alize Cornet of France, cruised past qualifier Anastasija Sevastova of Latvia 6-1, 6-3.
Cornet recovered from her nervous breakdown in a Fed Cup tie against Italy last weekend, although she asked twice for her coach on the court against Sevastova.
"At 4-1 in the second set, I felt that she was playing a bit better,'' Cornet said. "I needed him to talk to me at that moment. If you let that kind of match last longer, you never know what can happen.''
Next up for Cornet will be former top-5 player Daniela Hantuchova of Slovakia, whom Cornet beat in the Australian Open third round last month. Hantuchova, twice a quarterfinalist in the Open GDF Suez, beat Ekaterina Makarova of Russia 6-2, 6-2.
Monica Niculescu of Romania and Frenchwomen Nathalie Dechy and Emilie Loit advanced to the second round, too. Niculescu beat Timea Bacsinszky of Switzerland 6-4, 6-3, Dechy edged Sorana Cirstea of Romania 6-2, 1-6, 6-2 and Loit eliminated Nicole Vaidisova of the Czech Republic 6-1, 6-4.
Vaidisova, a former top-10 player, was struggling to get back to her best. Tuesday's loss was the 19-year-old Czech's third first-round exit of the year.
World No. 1 Serena Williams will play Iveta Benesova of the Czech Republic on Wednesday while Jelena Jankovic of Serbia will face Francesca Schiavone of Italy.
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