Fifth seeds Paes and Stepanek hit 25 winners en route their 6-3 6-2 win over unseeded pair of India's Yuki Bhambri and New Zealand's Michael Venus in the pre-quarterfinal, which lasted 65 minutes.
In his mixed doubles first round match, Paes, partnering Hantuchova, got the better of Ajla Tomljanovic of Croatia and Australia's James Duckworth 7-5 4-6 10-7 in an hour and 26 minutes to set up a second round clash against Mahesh Bhupathi and Elena Vesnina.
However, seventh seeded India's Rohan Bopanna and his Pakistani partner Aisam-Ul-Haq Qureshi endured a shocking defeat against the 12th seeded duo of Treat Huey and Dominic Inglot to crash out of the men's doubles event.
In the men's doubles pre-quarterfinal, Paes ended the dream run of young compatriot Bhambri in just 65 minutes.
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Paes and Stepanek broke the serve of their opponents three times and served notice of their intent as they look to win the Australian Open title for the second time, following their victory in 2012 over Bryan brothers.
They had added a second Grand Slam championship to their collection at last year's US Open with victory over Alexander Peya and Bruno Soares.