Like Swiatek, Gauff also had issues with her serve in the first round as she committed seven double faults against Uzbekistan’s Kamilla Rakhimova.
Three of those came in the opening game, but Gauff said it was “smooth sailing from there”.
“I think I just erased that first game, and then after that I was better,” she said following the 6-2 6-3 win – her 75th Grand Slam match-win.
The two-time major singles winner added: “I tried not to put too much pressure on myself. I am only satisfied if I win, but I am proud of myself regardless of how I get on.”
Gauff, 21, will face Olga Danilovic next after the Serb defeated 45-year-old Venus Williams on Sunday.
American fourth seed Anisimova, meanwhile, needed just 60 minutes to beat Switzerland’s Simona Waltert 6-3 6-2 and underline her title credentials.
After runs to the US Open and Wimbledon final last season, Anisimova is one of the favourites go all the way in Melbourne, but she insisted she was focusing on improving as a player rather than a third successive Grand Slam final.
Fellow American Pegula, the sixth seed, also impressed with a 66-minute 6-2 6-1 victory over Russian Anastasia Zakharova.
Russian teenager Mirra Andreeva – another title contender – found herself a set down in her first-round tie against Croatia’s Donna Vekic, but the eighth seed rallied in set two and won the decider 6-0, winning 25 of the 32 points on offer.