Kontaveit worked hard to secure an impressive 6-2, 4-6, 6-4 victory in over two hours. It was a match Kontaveit needed to win in order to remain in the competition after losing to Elise Mertens in her opening pool game on Tuesday.
World No.14 Georges now needs to beat Merterns in straight sets on Wednesday to have any hope of advancing from the Rose group and reaching the semifinals.
Elsewhere Madison Keys won her opening match of the competition with a comprehensive 6-2, 6-4 victory over No.1 seed Daria Kasatkina in the Azalea group. Keys can qualify for the semifinals on Friday if she beats home favourite Wang Qiang, who lost to Kasatkina on Tuesday.
The Elite Trophy features the world's top 12 singles players who did not qualify for the WTA Finals last week.
The round-robin format will see the winners from each of the four groups advance to the semifinals.