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 (21) of Italy also advanced by defeating Australia's Philip Sekulic (20) 7-6, 6-7, 6-0.
Other winners included Grigor Dimitrov (32), who saw off Juan Pablo Varillas (27) 6-3, 6-4, Arthur Rinderknech (28) and Miomir Kecmanovic (24).
Zhuhai Championships
Great Britain's Cameron Norrie (28) topped Australia's Marc Polmans (26) 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 (23) defeated France's Alexandre Muller (26) 6-1, 2-6, 6-3, Tomas Martin Etcheverry (25) of Argentina upended the Czech Republic's Dalibor Svrcina (20) 6-2, 6-3, and Russia's Aslan Karatsev (30) edged Great Britain's Andy Murray (36) 4-6, 6-3, 6-2.
