In a match littered with service breaks, Radwanska battled into the third round with a 6-4 6-3 triumph in one hour and 24 minutes.
Watson failed to capitalise on two break points in the opening game and quickly found herself 3-0 behind in the first set.
And despite rallying to 4-4, she failed to build on the momentum, eventually losing the set 6-4.
Both players struggled for rhythm in a second set which saw seven consecutive breaks, but it was Radwanska who prevailed to set up a clash with Elena Vesnina or Annika Beck.
"I'm just very happy I could play quite well today and win in two sets," the Pole told reporters a day after her 25th birthday.
"Today was very windy and we were both struggling with the serve, so I'm happy I could make it through."
Davis broke serve seven times en route to 6-0 7-6(2) victory as she pounced on the 2012 champion's second serve. Azarenka, who was hampered by a foot injury, did not help her cause with 12 double faults.
It is the first time 20-year-old Davis, ranked 66th in the world, has beaten an opponent inside the top 20.
The top seeds received a first round bye and most of the big names will not make their first appearances until the weekend.
Wozniacki broke Jovanovski five times as she preyed on her Serbian opponent's weak second serve.
Top seed Li Na will start her campaign on Saturday against Chinese compatriot Zheng Jie.
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- Elena Vesnina