The Serbian brushed aside Cristian Garin in his opener and did likewise against Van De Zandschulp, prevailing in just 71 minutes in the Kazakh capital.
Djokovic has won six matches in a row and is seeking back-to-back titles after triumphing at the Tel Aviv Open last week – his 89th ATP singles crown.
He reeled off five games in a row in the second set to advance 6-3 6-1, with that his 30th tour-level victory of the year.
As Djokovic nears the end of a rather mixed season, the 21-time grand slam winner is in the mood to add further titles to his collection.
"My season is different from any other so don't look at my season in comparison to others," he said in his post-match interview.
"I haven't played for over three months before Israel so I definitely am physically and mentally motivated to do well."
Awaiting Djokovic in the quarter-final is Karen Khachanov, who eliminated Marin Cilic with a 2-6 6-3 6-3 comeback victory.
World number four Daniil Medvedev also advanced thanks to a 6-3 6-2 win over Emil Ruusuvuori, while Hubert Hurkacz beat Alexander Bublik 6-4 6-4.
At the Japan Open, third seed Taylor Fritz defeated Hiroki Moriya in three sets and will now face fifth seed Nick Kyrgios, who recovered to beat Kamil Majchrzak 3-6 6-2 6-2.
Denis Shapovalov, looking to build on his run to the Seoul Open final, got the better of home hope Rio Noguchi with a 6-3 6-1 win to advance to the quarter-finals.