The seventh seeded Indo-Czech team brushed aside the spirited challenge of unseeded British pair 6-4 7-5 in one hour and nine minutes in the second round.
Bopanna and Hlavackova broke their rivals one time in each set to record the win.
They will now fight it out with Russia's Mikhail Elgin and Australian Anastasia Rodionova for a place in the quarterfinals.
Meanwhile, Sania Mirza and Horia Tecau, seeded sixth, were leading 4-3 in the opening set of their mixed doubles second round contest against Mate Pavic and Bojana Jovanovski when rain forced suspension of play.
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