superb display of deep, accurate groundstrokes to seal victory in just over an hour.
Second-seeded Sharapova, champion here in 2006, broke Schiavone's serve three times in each set and will next face Carla Suarez Navarro, a 6-1 6-1 winner over fellow Spaniard Silvia Soler Espinosa.
Svetlana Kuznetsova also advanced to the third round, recovering from a poor start after discarding her long tights to upset 18th-seeded Serb Jelena Jankovic 0-6 6-2 7-5.
"I was pretty bad in the first set, but I managed to change it," Kuznetsova, a losing finalist here in 2007 and 2008, told reporters. "So I'm pretty happy with it, but I have to change the start."
The Russian, who was sidelined by a right knee injury for six months last year after losing in the first round of Wimbledon, did not blame her tights for her slow start but said she felt much more comfortable without them.
"Even though I tried to play in long tight pants in the first set, I cannot," the 27-year-old smiled. "I can practise in them without a problem, but when I have matches I always start to lose and I have to take them off.
"They are very tight, so they were kind of pushing my stomach and I was not so much comfortable. It was not bothering me like so much, but after I took them off I was feeling a little bit looser. It's a little bit funny."
Kuznetsova, whose world ranking has plummeted to 46th from a career-high second, will next face ninth seed Marion Bartoli of France who brushed aside South Africa's Chanelle Scheepers 6-3 6-3.
Earlier, Wimbledon champion Petra Kvitova survived 10 double faults and an erratic second set performance to beat Olga Govortsova of Belarus 6-4 3-6 6-3 in the first match of the day on the second Stadium Court.
The big-serving Czech, seeded fifth in the elite WTA event at the Indian Wells Tennis Garden, sealed victory after a marathon encounter lasting two hours and 21 minutes when a Govortsova forehand sailed wide.
Kvitova, who clinched her 10th WTA singles title in Dubai last month, will next face Ukrainian Lesia Tsurenko, who beat Kazakhstan's Yaroslava Shvedova 6-1 3-6 6-1.
In other matches, sixth-seeded Italian Sara Errani battled past Spaniard Lourdes Dominguez Lino 6-2 3-6 6-3 and 12th seed Dominika Cibulkova of Slovakia beat Romania's Simona Halep 7-5 7-6.
Later in the day, third-seeded Agnieszka Radwanska of Poland was scheduled to take on American teenager Maria Sanchez.
Australian Open champion Victoria Azarenka of Belarus, who demolished Sharapova 6-2 6-3 in last year's BNP Paribas Open final, will launch her title defence against Slovakia's Daniela Hantuchova in the second round.