The India-Czech Republic duo will take on the Bryan brothers, Bob and Mike, who beat Australia's Carsten Ball and Chris Guccione 6-4, 7-6 (7-2), in the next round.
After winning the first set rather easily, Paes-Dlouhy took a 2-0 lead in the second, but were broken in the fourth and 12th games as Moodie and Norman took the tie to the decider.
Paes and Dlouhy broke immediately in the third set and raced away to a 4-1 lead after breaking again in the fifth game.
However, serving for the set, they were broken in the eighth game. But after the South Africa-Belgium pair held in the ninth, Paes ensured there were no further hiccups by serving out the next game for set and match.
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