Kasatkina, 28th in the world rankings, beat Aliaksandra Sasnovich 6-4 6-3.
Kasatkina will face Romanian Begu, who saved a match point in the third set before beating Belarusian qualifier Vera Lapko 6-3 4-6 7-5.
After losing the first two games of the match, seventh seed Goerges, the highest-ranked player left in the tournament, defeated Lesia Tsurenko 6-3 6-4.
She will next meet Russian Vikhlyantseva who upset Frenchwoman Alize Cornet 4-6 7-5 6-3.
(Reporting by Gabrielle Tétrault-Farber, editing by Ed Osmond)
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