Top seed Sakkari beat Anastasia Potapova 6-4 6-4 to advance to the last 16, while seventh seed Ostapenko was a 6-1 6-4 winner against wildcard Wang Xinyu.
Kvitova, the 2018 champion, took just an hour and nine minutes to see off qualifier Jule Niemeier 6-2 6-1, with the sixth seed and two-time former Wimbledon winner making a positive first appearance since being eliminated from the Australian Open in round one.
Sakkari, Ostapenko and Kvitova followed fellow seeds Anett Kontaveit, Belinda Bencic and Elise Mertens - winners on Monday - in avoiding an early exit.
However, number four seed and home favourite Anastasia Pavlyuchenkova was forced to withdraw from the competition on Tuesday due to a knee problem.
Bernarda Pera replaced the Russian in the main draw and fell 7-6 (7-3) 6-4 to Jaqueline Cristian, who will next face Aliaksandra Sasnovich.
That is after Sasnovich held firm to beat Magda Linette 7-5 4-6 6-4 in a tight contest, with a decisive break in the fifth game of the third set seeing her through.