Azarenka overcame a knee injury during his Opening Day match against Portugal's Maria Joao Koehler, eventually routing her 6-1, 6-2 after enduring a fall that prompted her to scream in pain.
The former world No. 1, however, recovered nicely and had little trouble finishing off Koehler. The Belarusian star now goes on to face Flavia Pennetta of Italy after winning for the 19th time in 21 Grand Slam matches.
Russia's Sharapova, meanwhile, downed Kristina Mladenovic of France 7-6 (7-5), 6-3 on Center Court of the All England Club. Her next opponent will be Portuguese Michelle Larcher de Brito as she seeks her second Wimbledon title.
The news wasn't as good for Italy's Errani, however, who was upended by 19-year-old Monica Puig of Puerto Rico 6-3, 6-2.
Other seeded winners Monday included No. 8 Petra Kvitova of the Czech Republic, No. 9 Caroline Wozniacki of Denmark, No. 17 American Sloane Stephens, No. 15 Marion Bartoli of France, No. 19 Carla Suarez Navarro and No. 20 Belgian Kirsten Flipkens.
Czech Eva Birnerova upended 26th-seeded American Varvara Lepchenko.