Kartal, 23, leads the first set by chance with the only resting point for the fourth seed.
Andreeva stressed the reaction in the second set as Qatar struggled with 60% first-time service percentage, breaking twice.
But Kartal, the world’s No. 81, showed her resilience in the third set as Andreeva failed to build on her momentum.
After failing to reach the match point, Qatar seized a second chance and Andreeva drove wide.
Kartal fell in just four rounds in the last eight matches in Beijing, where she would face world No. 27 Linda Noskova to reach the semi-finals.
