World number three Roger Federer was briefly hauled off court while the roof was closed on Rod Laver Arena, before returning to complete a routine 6-3, 6-2, 6-2 victory over Steve Johnson.
Play also continued under the retractable roofs of Margaret Court Arena and Melbourne Arena, but was impossible elsewhere as rain persisted.
More rain is forecast for Tuesday, threatening further disruption at a tournament more often associated with extreme heat at the height of the Australian summer.
Officials said play would resume on the outside courts once the rain stops, and would continue into the evening under floodlights to complete Monday’s schedule of 64 matches.
Defending champion Osaka avoided the downpour as she dismissed Marie Bouzkova of the Czech Republic 6-2, 6-4 in 80 minutes, breaking a net fitting with one powerful serve in the process.
“It was really tough for me trying to control my nerves,” said Osaka. “It’s tough to play someone you’ve never played before in the first round of a Grand Slam.”
Williams, on the hunt for a record-equalling 24th Grand Slam title, won the first set against Anastasia Potapova in just 19 minutes as she cruised to a 6-0, 6-3 win in less than an hour.
“I felt like I started out really well. Played really strong in the first set and just building on that,” Williams said.
“I feel like I can still improve and get better throughout this tournament, for sure. This is a good stepping stone for right now.”
Later on Margaret Court Arena, Williams’s elder sister, 39-year-old Venus, faces rising star Coco Gauff, 15, who is looking to reprise her upset of the seven-time Grand Slam winner at Wimbledon last year.
Australian world number one Ashleigh Barty, fresh from winning the Adelaide International on Saturday, plays Ukraine’s Lesia Tsurenko on Rod Laver Arena.
Player anger over smog dominated the final days before the tournament, which is taking place after deadly bushfires ravaged large parts of Australia.
Slovenia’s Dalila Jakupovic was forced to retire from her qualifier after a coughing fit, while Britain’s Liam Broady claimed “multiple” players needed asthma medication.
“After I practised indoors I felt really bad with my lungs,” said Greece’s Stefanos Tsitsipas, the world number six.
“I was coughing a lot had troubles breathing for a couple of hours.”
Federer was one of the competitors who criticised a lack of communication from tournament officials, who were forced to suspend practice and delay some qualifying matches last week.
Tournament officials are closely monitoring pollution and will halt play and close the three main stadiums’ roofs if particulate matter suspended in the air hits PM2.5 200.
In other results, Canadian 13th seed Denis Shapovalov argued with the umpire in a tempestuous defeat as he lost in four sets to Marton Fucsovics after earning a code violation for throwing his racquet.
Croatia’s 25th seed Borna Coric was also an early casualty as he went down in three sets to experienced American Sam Querrey.
But former champion Caroline Wozniacki, playing her last tournament before retiring, safely reached the second round as she beat Kristie Ahn 6-1, 6-3.