Fifth seeds Paes and Stepanek hit 25 winners en route their 6-3 6-2 win over India's Bhambri and his Kiwi companion Michael Venus in the pre-quarterfinal, which lasted 65 minutes. (Agencies)
The Indo-Czech pair broke the serve of their opponents three times to book their berth in the last-eight stage of the tournament.
Paes and Stepanek are looking 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.
Meanwhile, the seventh seeded pair of 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.
The Filipino-British pair of Huey and Inglot scored an upset 6-4 7-6(1) win over last week's Sydney finalists Bopanna and Qureshi in the top half of the draw.
Huey-Inglot sent down 10 aces and hit 35 winners to set up a potential quarter-final duel with top seeds Bob Bryan and Mike Bryan.
In the junior girls' singles event, India's Dhruthi Tatachar Venugopal lost 3-6 3-6 to Germany's Katie Boulter.
In the junior boys' doubles first round, the eighth seeded pair of India's Sumit Ngal and Korea's Hyeon Chung went down to the Spanish duo of Martinez Portero and Munar Clar 3-6 6-7.
Fifth seeds Paes and Stepanek hit 25 winners en route their 6-3 6-2 win over India's Bhambri and his Kiwi companion Michael Venus in the pre-quarterfinal, which lasted 65 minutes.