The reward is a match-up with older sister Venus, after the eighth seed saw off Sorana Cirstea 6-3 6-4.
Williams will now face older sister Venus, whom she beat in the Australian Open decider at Melbourne Park previous year. "It was looking scary in the second set so it was good to pull it out", Venus said.
"It was unbelievable. I haven't played more than a year, my daughter was born, and now I need to go to her".
She is not yet in prime form, not yet in prime shape, but in her first two matches she has proved she still has staying power, coming to the net effectively, something she appears to be emphasizing in this comeback, and playing the crucial points with trademark intensity.
"She's playing really well and just honing her game. It's just afterwards I don't like it as much".More news: Warner v de Kock spat exposes Australian sledging hypocrisy... again
"Obviously I wish it was anybody else in the draw, literally anybody, but that's okay".
Serena Williams owns a 17-11 record against Venus Williams, having taken the most recent matchup in the final at Melbourne a year ago.
Fourth seed Elina Svitolina of Ukraine beat Germany's Mona Barthel 6-4, 6-3 while compatriot and No. 12 seed Julia Georges defeated Russia's Natalia Vikhlyantseva 6-4, 6-1.
In other matches on Saturday, defending champion Elena Vesnina was made to work against American CiCi Bellis, losing the first set before coming back strongly to win 2-6, 6-1, 6-1.
In the men's competition, Swiss tennis legend Roger Federer won his first set against Federico Delbonis of Argentina.More news: Honor 7C With Snapdragon 450 SoC and Android Oreo Goes Official
Venus and Serena have faced each other 28 times in their legendary careers and they are now meeting in Serena's first tournament back on the WTA Tour since she beat Venus 6-4, 6-4 in the 2017 Australian Open final.
Saturday night rain caused the suspension of one women's singles match and the postponement of another.
The highest-seeded player to fall was No. 15 Madison Keys, who lost 6-3, 7-6 (1) to fellow American Danielle Collins.
Fifth-seeded Dominic Thiem fired 10 aces to beat Stefanos Tsitsipas of Greece 6-2, 3-6, 6-3; No. 7 Kevin Anderson defeated Evgeny Donskoy 7-5, 6-4; and No. 12 Tomas Berdych rolled past German Maximilian Marterer 6-1, 6-4 in just over an hour. However, in the time when play was possible Spaniard Fernando Verdasco used his powerful serve to upset third seed Grigor Dimitrov 7-6 (4), 4-6, 6-3.More news: Arsenal's Win Over Watford Is Important, Says Arsene Wenger