The 20-year-old Scot is on the comeback trail after a summer wrecked by a bad wrist injury and he served up a typically erratic performance against Bjorkman.
He next takes on 31-year-old Korean Lee Hyung-taik, who upset Guillermo Canas 7-5, 7-5, 6-3.
No. 4 seed Nikolay Davydenko, beset by questions about illegal tennis gambling patterns, focused to breeze past Germany's Nicolas Kiefer 6-2, 6-2, 6-2. Davydenko next plays Spain's Nicolas Almagro.
Tommy Haas of Germany and Sebastien Grosjean of France also won. They will play each other for a slot in the final 16.
The bottom half of the women's draw also reached the third round stage with no major upsets.
Sharapova apart, there were wins for three Russian threats -- fourth seed Svetlana Kuznetsova, sixth seed Anna Chavetadze and seventh seed Nadia Petrova.
Martina Hingis also moved on with a 6-2, 7-5 win over France's Pauline Parmentier.
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