Williams, meanwhile, headed into the opening slam of the season in high spirits following her pre-tournament title success in Auckland.
Williams looked frustrated during the second set, when she committed 17 of her 28 unforced errors, before celebrating her 400th grand slam singles match with a win.
Looking at Serena Williams, the American superstar was dominant during the opening round.
"I still have a lot more to, I guess, become like one of those 'big names, '" she continued, making air quotes with her fingers. "So that's really tough too", she said.
"I think I made so many errors in a row and I just had to battle through that, through my own internal problems".
Finding the range right from the start, Serena earned a break at 15 in the first game and fired a service victor in game two to confirm the break and cement the advantage.
She said she felt a little ring-rust given she didn't qualify for the WTA championships and needed to work out how to get the game back on her terms.
"I hadn't been able to win as a mum, so it was nice to finally be able to win a tournament with a two-year-old now", Williams said. "It's different. It's better than just going to the gym every day". I hadn't done that since September.
Serena will next face Wang Qiang, while Gauff is also into the third round after a gutsy win over Sorana Cirstea and takes on Naomi Osaka in the third round.
Switzerland's Roger Federer makes a backhand return to Serbia's Filip Krajinovic. "But for today, my mission was to win", said Gauff, whose good friend and doubles partner, 18-year-old American qualifier Caty McNally, upset 2011 U.S. Open champion Sam Stosur of Australia 6-1, 6-4 at night.
Twenty-three time Grand Slam victor Serena Williams.
"I always dance. I'm not the best dancer, but I love dance", Williams told reporters.