Poland’s world number one Swiatek, who is aiming to win the title in Rome for a third time, dominated American Bernarda Pera on the way to a 6-0 6-2 win in one hour 17 minutes.
American Gauff was a 6-3 6-3 winner against Poland’s Magdalena Frech in one hour 28 minutes.
Japan’s Osaka completed a 6-3 6-2 win against world number 20 Marta Kostyuk, of Ukraine.
The straightforward rain-interrupted win in one hour 11 minutes gave Osaka her first career victory against a top 20 player on clay.
She will face world number 11 Daria Kasatkina in the third round.
“I’m going into this match knowing that she has way more experience than me on clay,” said Osaka. “I feel like I’m still a baby giraffe, like I’m trying to learn my steps.
“I’m just extremely humble. I want to learn from the match I played today and hopefully apply that.”
