Former world number one Azarenka, playing her first Grand Slam tournament since giving birth to first child Leo in December, knocked out Russian 15th seed Elena Vesnina.
The 2012 and 2013 Australian Open champion, in her fourth match back, won 6-3, 6-3, taking her record over the Russian to 8-0 without dropping a set.
The Belarusian, 27, could become the first mother to win the Wimbledon singles since Evonne Goolagong Cawley in 1980.
"Travelling is a little stressful for me, travelling with a baby. I want to make sure that everything is okay. But overall, it's just great. It's really fun that I'm able to go home and spend time with my son," she said.
"Overall, I felt like I played really good on the important moments, which gave me opportunities to win comfortably," she said of the match.
Azarenka next faces British wildcard Heather Watson who also upset the form book. The world number 102 knocked out Latvian 18th seed Anastasija Sevastova in straight sets.
"I think Vika is an incredible player and person," Watson said of Azarenka.
"I know she has only just come back, but I expect her to be playing really well because I know she will have put in the hours and the work."
Murray facing unpredictable Brown
Men's defending champion Murray was due to face the unpredictable serve-volleyer Dustin Brown on Centre Court, before Nadal steps out into the 15,000-seater arena.
Nadal, the 2008 and 2010 champion, is looking to reach the third round for only the second time since he finished runner-up in 2011 when he faces fellow left-hander Donald Young of the United States.
Fresh from winning a historic 10th French Open, Nadal is looking bury memories of the 2015 second round defeat to Brown, his last appearance at the All England Club.
Seventh seed Marin Cilic, the 2014 US Open champion, is in form on grass having reached the Queen's final. The Croat faces Germany's Florian Mayer for a spot in the third round.
Spanish 18th seed Roberto Bautista Agut was the first man through to round three with a four-set victory over German qualifier Peter Gojowczyk.
He next faces either Japanese ninth seed Kei Nishikori or Ukrainian qualifier Sergiy Stakhovsky.
France's Jo-Wilfried Tsonga, the 12th seed, made short work of Italian qualifier Simone Bolelli in the sizzling heat, winning in straight sets.
Fans shielded their faces from the sun with towels, programmes and napkins as temperatures reached 29 degrees Celsius (84 degrees Fahrenheit).
Williams, Kvitova in the wings
Five-time champion Williams is trying to concentrate on tennis against the backdrop of the impending legal action she faces back home in the United States over a fatal car crash.
The 37-year-old veteran, devastated by the incident, broke down in tears on Monday when talking about the crash. She faces Chinese rising star Wang Qiang on Court One.
Two-time champion Petra Kvitova, the bookmakers' favourite for the title, continues her comeback from six months out recovering from a horrific knife attack in her home with a match against Madison Brengle of the United States.
Eighth seed Dominika Cibulkova saw off Jennifer Brady of the United States in straight sets. The 2014 Australian Open finalist faces 27th seed Ana Konjuh of Croatia in round three.
In Wednesday's later matches, French Open champion Jelena Ostapenko faces Canada's Francoise Abanda, while second seed Simona Halep takes on Beatriz Haddad Maia of Brazil.