Winger Kluivert came on as a 39th-minute substitute for the injured Amin Younes with the game goalless.
Lasse Schone's penalty put Ajax ahead and Hakim Ziyech doubled the lead.
Nicolai Brock-Madsen pulled a goal back, but Ziyech sealed a victory that ensures Ajax stay five points behind leaders Feyenoord, who beat Roda 2-0.
Kluivert Sr won two Dutch Eredivisie titles and the Champions League during three years at Ajax between 1994 and 1997.
He went on to play for AC Milan, Barcelona, Newcastle United, Valencia, PSV Eindhoven and Lille and scored 40 goals in 79 appearances for his country.
The 40-year-old is now director of football at French champions Paris St-Germain.
Match ends, PEC Zwolle 1, Ajax 3.
Second Half ends, PEC Zwolle 1, Ajax 3.
Attempt blocked. Mustafa Saymak (PEC Zwolle) right footed shot from outside the box is blocked. Assisted by Stef Nijland.
Davy Klaassen (Ajax) wins a free kick on the right wing.
Foul by Ted van de Pavert (PEC Zwolle).
Offside, Ajax. Donny van de Beek tries a through ball, but Anwar El Ghazi is caught offside.
Attempt missed. Stef Nijland (PEC Zwolle) right footed shot from outside the box is high and wide to the right. Assisted by Hachim Mastour.
Foul by Joël Veltman (Ajax).
Hachim Mastour (PEC Zwolle) wins a free kick in the defensive half.
Attempt missed. Stef Nijland (PEC Zwolle) right footed shot from outside the box is close, but misses to the left. Assisted by Queensy Menig.
Offside, PEC Zwolle. Stef Nijland tries a through ball, but Nicolai Brock-Madsen is caught offside.
Attempt saved. Stef Nijland (PEC Zwolle) right footed shot from outside the box is saved in the centre of the goal.
Substitution, Ajax. Donny van de Beek replaces Lasse Schöne.
Attempt missed. Anwar El Ghazi (Ajax) right footed shot from the right side of the six yard box is close, but misses to the right. Assisted by Joël Veltman.
Substitution, PEC Zwolle. Stef Nijland replaces Wouter Marinus.
Attempt missed. Mustafa Saymak (PEC Zwolle) right footed shot from outside the box is too high. Assisted by Queensy Menig.
Goal!  PEC Zwolle 1, Ajax 3. Hakim Ziyech (Ajax) left footed shot from a difficult angle on the left to the centre of the goal. Assisted by Daley Sinkgraven following a fast break.
Foul by Lasse Schöne (Ajax).
Wouter Marinus (PEC Zwolle) wins a free kick in the attacking half.
Attempt missed. Hakim Ziyech (Ajax) left footed shot from outside the box is high and wide to the right. Assisted by Joël Veltman.
Offside, PEC Zwolle. Ryan Thomas tries a through ball, but Nicolai Brock-Madsen is caught offside.
Attempt saved. Nicolai Brock-Madsen (PEC Zwolle) right footed shot from the centre of the box is saved in the centre of the goal. Assisted by Dirk Marcellis with a cross.
Attempt saved. Anwar El Ghazi (Ajax) left footed shot from outside the box is saved in the bottom right corner. Assisted by Davy Klaassen.
Attempt blocked. Kasper Dolberg (Ajax) right footed shot from outside the box is blocked. Assisted by Nick Viergever.
Kasper Dolberg (Ajax) wins a free kick in the attacking half.
Foul by Ted van de Pavert (PEC Zwolle).
Justin Kluivert (Ajax) wins a free kick in the defensive half.
Foul by Dirk Marcellis (PEC Zwolle).
Foul by Joël Veltman (Ajax).
Ryan Thomas (PEC Zwolle) wins a free kick in the defensive half.
Goal!  PEC Zwolle 1, Ajax 2. Nicolai Brock-Madsen (PEC Zwolle) header from the centre of the box to the top left corner. Assisted by Queensy Menig with a cross.
Attempt saved. Philippe Sandler (PEC Zwolle) right footed shot from more than 35 yards is saved in the centre of the goal. Assisted by Mustafa Saymak.
Substitution, PEC Zwolle. Ryan Thomas replaces Josef Kvída.
Offside, PEC Zwolle. Philippe Sandler tries a through ball, but Queensy Menig is caught offside.
Attempt missed. Queensy Menig (PEC Zwolle) right footed shot from the right side of the box is too high. Assisted by Nicolai Brock-Madsen.
Foul by Justin Kluivert (Ajax).
Hachim Mastour (PEC Zwolle) wins a free kick in the defensive half.
Attempt saved. Nicolai Brock-Madsen (PEC Zwolle) header from the centre of the box is saved in the centre of the goal. Assisted by Wouter Marinus.
Daley Sinkgraven (Ajax) is shown the yellow card for a bad foul.
Foul by Daley Sinkgraven (Ajax).
Justin Kluivert, the 17-year-old son of former Netherlands striker Patrick, made his Ajax debut in a comfortable win at PEC Zwolle on Sunday.