The duo defeated Croatian Ivan Dodig and American Austin Krajicek 7-6, 6-2 in their round-of-16 match on Monday.
Up next for Zieliński and Nys at the clay-court tournament in the Italian capital is the American duo of Mackenzie McDonald and Frances Tiafoe.
Meanwhile, Poland's world No. 1 Iga Świątek is set to take on Croatia's Donna Vekić for a place in the last eight of the women's singles at the Italian Open.
Świątek is the last Pole still standing in the singles at this year's Italian Open, which runs until Sunday.
Poland's world No. 19 Magda Linette has crashed out of the tournament after being eliminated by Brazilian Beatriz Haddad Maia in the third round of the women's singles.
On the men's side, Poland’s Hubert Hurkacz bowed out of the tournament after being bested by American J.J. Wolf 3-6, 4-6 in the second round of the singles in Rome.
(gs)
Source: IAR, PAP