Former world number eight Stepanek came from a set down to beat Tatsuma Ito 6-7(5) 7-6(5) 6-1 7-5 before Rosol survived a mid-match meltdown to overcome 21-year-old Taro Daniel 6-4 6-4 3-6 4-6 6-2 in the second rubber.
Stepanek, 35, fired down 18 aces to quell the challenge of his 146th ranked opponent in three hours, 53 minutes at the Ariake Coliseum after going down in a tight first set tiebreak.
"The second set was bit scary, he got to a set point. But I played an unbelievable volley on the set point which I think was a very important moment of the match," Stepanek said. "I was able to sneak out from the tiebreak.
"Sometimes you have days when you are not playing your best tennis but you always have to find a way through. Today was a day like that and I am very happy that I was able to bring the first point to our locker room."
Rosol, ranked 40, twice broke Daniel's serve in each of the first two sets to surge ahead but the Japanese debutant fought back, bagging the next two sets to even the rubber out at 2-2.
The Czech got an early break in the final set and also converted his second break point to seal the match out after three hours and 12 minutes.
Ito and Daniel are bearing the weight of Japan's hopes of prgressing after world number 18 Kei Nishikori withdrew with a groin strain and Go Soeda was laid low by a fever.
Japan, Davis Cup runners-up in 1921, are playing their first World Group quarter-final tie.
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