As she did in her first round match, 23-year-old Sun, who was born in Te Anau, lost the opening game against Starodubtseva 4-6 before taking the second 6-3 on Wednesday.
Sun then finished off the match, that took just over two hours to complete, winning the third set 6-2.
She will play China's Lin Zhu in the third round. Zhu has had straight set wins in her first two rounds.
Before this week Sun had never won a match at a Grand Slam.
Women's doubles second seeds New Zealand's Erin Routliffe and her Canadian partner Gaby Dabrowski defeated Mirra Andreeva and Anastasia Potapova 6-2 7-5 to move on the to second round where they will meet either a Brazilian duo or Russian pairing.
New Zealand men Michael Venus and Marcus Daniell did not get to play their opening round doubles matches due to rain delays.
Sun and Routliffe will team up for New Zealand at the Paris Olympics.
Tokyo bronze medalists Venus and Daniell missed Olympic selection this year.
