Top seed Switaek, seeking a fifth consecutive title, breezed past Elena-Gabriela Ruse 6-3 6-0 as she became the first female player to win 24 consecutive matches since Serena Williams in 2015.
Badosa followed suit by dispatching Aliaksandra Sasnovich 6-2 6-1 in just an hour and 17 minutes, teeing up a third-round clash with Daria Kasatkina, who beat Leylah Fernandez 4-6 6-3 7-6 (7-2).
Aryna Sabalenka, the third favourite in Italy, also made light work of Zhang Shuai to triumph 6-2 6-0, while fourth seed Maria Sakkari cruised past Madrid Open semi-finalist Ekaterina Alexandrova 6-3 6-2.
Sakkari will next face a Rome rematch with Coco Gauff, who defeated compatriot Madison Brengle 6-2 6-4. Teenager Gauff's sole win against Sakkari came at this tournament in the second round last year.
Yulia Putintseva came from behind to beat Garbine Muguruza 3-7 7-6 (7-4) 6-1, and will next meet Ons Jabeur, who made it eight successive main-draw wins by defeating Ajla Tomljanovic 7-5 6-2.
Seventh seed Danielle Collins also battled past former world number one Simona Halep 7-6 (7-1) 6-3.
There was no such joy for fifth seed Anett Kontaveit as she crashed out to Petra Martic in straight sets, while Jil Teichmann upset former Rome champion Karolina Pliskova in a three-set thriller.
Belinda Bencic, who was ranked as 12th seed, was another early casualty as she fell to a 7-6 (7-4) 6-1 defeat against American Amanda Anisimova.
Meanwhile, Jessica Pegula secured her third-round spot after Anhelina Kalinina withdrew due to injury, with Victoria Azarenka earning a straightforward 6-2 6-4 victory over Camila Osorio to get a shot at favourite Swiatek in the next round.