In a ruthless start to her quest for a historic 24th career major, Williams cast sentiment aside to sweep past good friend, neighbour and fellow tennis mum Tatjana Maria 6-0 6-2 on Tuesday.

The match was the American superstar's first on RLA since beating older sister Venus, also a Tuesday winner, while pregnant in the 2017 final.

"It was nice to be back out there. Last time I was out there was a great moment. Especially internally for me, it was even a better moment," Williams said.

"It was literally the best win of my career."

But, as much as she savoured her return, Williams didn't want to leave 16-month-old daughter Olympia waiting much more than the 49 minutes it took to see off the German.

"Spending a lot of time with my daughter, that's the priority for me," said the former world No.1.

"As a working mum, I feel guilty. And I understand that that's normal, but these are years I'll never get back.

"I just try to spend every moment that I can when I'm not working with her."

Decked out in a new bottle-green fitted playsuit which she described as a "Serena-tard", Williams was also playing her first grand slam match since last year's US Open final meltdown.

But after being penalised for illegal coaching before losing to Naomi Osaka in New York, Williams was in no mood to discuss the prospect of on-court coaching being introduced when asked in her post-match press conference.

Nor was the 37-year-old prematurely planning celebrations for possibly matching Margaret Court's all-time record 24 grand slam singles titles after falling a win short both at Flushing Meadows and Wimbledon in 2018.

"I have been going for the record for what seems like forever now, so it doesn't feel any different," Williams said.

The seven-times Open champion will resume on Thursday against Eugenie Bouchard after the Canadian ousted Chinese wildcard Peng Shuai 6-2 6-1.

World No.1 Simona Halep did it tougher against Estonian slugger Kaia Kanepi, winning 6-7 (2-7) 6-4 6-2.

World No.8 Karolina Pliskova also advanced with a 6-3 6-2 victory over fellow Czech Karolina Muchova, and will next play American Madison Brengle, a 6-4 6-0 winner over Japanese qualifier Misaki Doi.

Two-time major winner and former world No.1 Garbine Muguruza beat China's Saisai Zheng 6-2 6-3.

Sixth-seeded Ukrainian Elena Svitolina, Belgian 12th seed Elise Mertens, Latvian No.13 seed Anastasija Sevastova, American 17th seed Madison Keys and Venus Williams also won.

Osaka, the fourth seed, was the last woman to book her ticket into the second round, defeating Magda Linette 6-4 6-2 to tee up a meeting with Tamara Zidansek, the Slovenian vanquisher of Daria Gavrilova.

Australian Associated Press