Azarenka, the No. 2 seed and Australian Open champion, defeated Irina Falconi 6-1, 6-4. Azarenka, who had nine double faults, dropped serve three times while posting six breaks against Falconi.
Kvitova, seeded fourth after winning at Wimbledon in 2011, overcame an early break for a 6-4, 6-4 win over Akgul Amanmuradova. The first service game break was the only one against Kvitova on the day as she ended up winning more than two-thirds of the points on serve.
The sixth-seeded Williams had just 12 unforced errors in a 6-2, 6-4 decision over Barbora Zahlavova Strycova. She piled up six breaks while winning 58 percent of the points off Zahlavova Strycova's serves.
Bartoli backed up her No. 9 seeding in a 6-2, 6-4 victory over Casey Dellacqua. Bartoli didn't lose her serve in the match.
Several matches were unable to be completed Tuesday because of a late rain storm. Those who will finish their first-round play Wednesday include former world No. 1 Caroline Wozniacki, the No. 7 seed, and French Open runner-up Sara Errani, who is seeded 10th.
No. 12-seeded Vera Zvonareva advanced with a 2-6, 7-6 (7-3), 6-4 win over Mona Bethel.
Kiki Bertens eliminated 19th-seeded Lucia Safarova 6-3, 6-0 in 50 minutes. The second set, in which Bertens held Safarova to 19 points, lasted just 19 minutes. Yanina Wickmayer defeated No. 32nd-seeded Svetlana Kuznetsova 6-2, 6-3 in the only other upset Tuesday.
Other seeded players moving into the second round Tuesday were 21st-seeded Roberta Vinci, No. 24 Francesca Schiavone, 25th-seeded Zheng Jie, No. 26 Anabel Medina Garrigues, No. 28 Christina McHale and 31st-seeded Anastasia Pavlyuchenkova.
Additional unseeded winners Tuesday included Varvara Lepchenko, Aleksandra Wozniak, Romina Oprandi, Melinda Czink, Mirjana Lucic, Kristyna Pliskova, Yaroslava Shvedova, Marina Erakovic, Galina Voskoboeva, Anne Keothavong, Elena Baltacha and Jana Cepelova.