Fourth-seed Dementieva, who is also the 2008 Beijing Olympics gold medalist, defeated unseeded Iveta Benesova of the Czech Republic in straight sets 6-4, 6-1 and will now meet with Australia's Samantha Stosur.
Her compatriot Kuznetsova (seeded 8) also enjoyed a quick straight sets victory in a match against unseeded Tatjana Malek of Germany. She will now meet Ukraine's Alona Bondarenko (31) in the third round.
Andreev, however, fought through a tough match against Latvia's Ernests Gulbis. After winning the first two sets, the No. 18 seed dropped the next two, but managed to recover in the fifth and final set for a 6-4, 6-4, 5-7, 3-6, 6-4 win.
He will now play ninth seed James Blake of the United States in the third round.
In all, seven Russian women and two men have reached the third round of the year's first Grand Slam.