The top seed in the tournament, Zverev survived a marathon match of two hours and 26 minutes by winning on return points at a 36% (37 of 102) to 25% (20 of 80) for Kotov.
Number two seed Lorenzo Musetti of Italy also advanced by defeating Australia's Philip Sekulic 7-6, 6-7, 6-0.
Other winners included Grigor Dimitrov, who saw off Juan Pablo Varillas 6-3, 6-4, Arthur Rinderknech and Miomir Kecmanovic.
Zhuhai Championships
Great Britain's Cameron Norrie topped Australia's Marc Polmans 6-0, 6-3 to reach the Zhuhai Championships quarter-finals.
Ranked number 17 in the ATP and seeded second in the tournament, Norrie coasted in 77 minutes by recording five aces and converting 75% (33 of 44) on service points.
Also on Saturday, Sebastian Korda defeated France's Alexandre Muller 6-1, 2-6, 6-3, Tomas Martin Etcheverry of Argentina upended the Czech Republic's Dalibor Svrcina 6-2, 6-3, and Russia's Aslan Karatsev edged Great Britain's Andy Murray 4-6, 6-3, 6-2.