The two Americans could meet in the semifinals.
A quarterfinalist in Melbourne in each of the past two years, Pegula needed just 59 minutes to get past the 161st-ranked Cristian, who was appearing in her third Grand Slam event.
Gauff, who reached her first Grand Slam singles final at the French Open last year, was given the honor of opening proceedings on Rod Laver Arena and dominated Siniakova in the first set.
The Czech player led 4-2 in the second set, but Gauff rallied and clinched victory on her seventh match point.
“I was not expecting to open the tournament on Rod Laver,” Gauff said. “I’m super honored that the tournament chose me and Katerina. I’m really pleased with myself. Katerina’s a fighter, I knew she was going to fight for every point. I just stayed strong mentally.”