Italy's Vinci made quick work of countrywoman Camila Giorgi 6-4, 6-2 just before heavy rains came down at the USTA Billie Jean King National Tennis Center, necessitating a 4-hour rain delay.
Qualifier Giorgi had gained the round of 16 with an upset of former world No. 1 Caroline Wozniacki Saturday.
Vinci's next foe will be another Italian, 31-year-old Flavia Pennetta, who racked up five service breaks in her 6-2, 7-6 (7-3) win over 21st-seeded Romanian Simona Halep Monday.
The 23-year-old Riske's match against Daniela Hantuchova was held up by rain in the second set, and the delay didn't help her as the Slovakian went on to finish her off 6-3, 5-7, 6-2.
Hantuchova reached her first U.S. Open quarterfinal in 11 years with the victory.