Sania defeated Kutuzova 6-4, 6-3 in one hour and 23 minutes to set up a tough quarter-final against second-seeded Frenchwoman Marion Bartoli.
Both Sania and Viktoriya served badly in the opening set, committing five double faults each.
But Sania managed to convert a crucial break point to go 1-0 up.
The Indian was more in control in the second set and she cut back on her service errors, saving three of the four break points she faced.
Viktoriya, however, continued to serve poorly and was broken thrice by the Indian, who was also helped by three double faults from the Ukrainian.
Sania goes into the match against world number 12 Bartoli with a 1-0 head-to-head record as the Hyderabadi had beaten her at the 2005 US Open.