The hosts were two up after seven minutes thanks to a Miralem Pjanic own goal and a Andrey Galabinov penalty.
But Dybala pulled one back from Pjanic's pass and then scored a penalty to level.
Juan Cuadrado put Juve ahead when he chested the down before cutting inside and curling home, and Dybala crashed home a fourth in injury time.
Match ends, Genoa 2, Juventus 4.
Second Half ends, Genoa 2, Juventus 4.
Gianluigi Buffon (Juventus) wins a free kick in the defensive half.
Foul by Gianluca Lapadula (Genoa).
Goal!  Genoa 2, Juventus 4. Paulo Dybala (Juventus) left footed shot from the right side of the box to the bottom right corner. Assisted by Gonzalo Higuaín.
Attempt missed. Gianluca Lapadula (Genoa) header from the centre of the box is high and wide to the right. Assisted by Raffaele Palladino with a cross.
Attempt saved. Raffaele Palladino (Genoa) left footed shot from the centre of the box is saved in the bottom left corner. Assisted by Andrea Bertolacci.
Gonzalo Higuaín (Juventus) wins a free kick in the defensive half.
Foul by Andrea Bertolacci (Genoa).
Substitution, Juventus. Rodrigo Bentancur replaces Miralem Pjanic.
Substitution, Genoa. Gianluca Lapadula replaces Goran Pandev.
Diego Laxalt (Genoa) is shown the yellow card for a bad foul.
Juan Cuadrado (Juventus) wins a free kick on the right wing.
Foul by Diego Laxalt (Genoa).
Foul by Alex Sandro (Juventus).
Goran Pandev (Genoa) wins a free kick in the defensive half.
Corner,  Juventus. Conceded by Davide Biraschi.
Attempt blocked. Miralem Pjanic (Juventus) right footed shot from outside the box is blocked.
Substitution, Juventus. Andrea Barzagli replaces Stephan Lichtsteiner.
Substitution, Genoa. Ricardo Centurión replaces Andrej Galabinov.
Attempt missed. Miguel Veloso (Genoa) right footed shot from outside the box is high and wide to the right.
Juan Cuadrado (Juventus) wins a free kick in the attacking half.
Foul by Santiago Gentiletti (Genoa).
Foul by Mario Mandzukic (Juventus).
Goran Pandev (Genoa) wins a free kick in the attacking half.
Attempt missed. Diego Laxalt (Genoa) left footed shot from outside the box is close, but misses to the right. Assisted by Miguel Veloso.
Juan Cuadrado (Juventus) is shown the yellow card for a bad foul.
Foul by Juan Cuadrado (Juventus).
Diego Laxalt (Genoa) wins a free kick on the left wing.
Goal!  Genoa 2, Juventus 3. Juan Cuadrado (Juventus) left footed shot from the centre of the box to the top left corner. Assisted by Mario Mandzukic.
Corner,  Juventus. Conceded by Davide Biraschi.
Attempt missed. Gonzalo Higuaín (Juventus) right footed shot from the centre of the box is too high. Assisted by Stephan Lichtsteiner with a cross.
Substitution, Juventus. Blaise Matuidi replaces Sami Khedira.
Offside, Genoa. Goran Pandev tries a through ball, but Andrej Galabinov is caught offside.
Foul by Giorgio Chiellini (Juventus).
Andrej Galabinov (Genoa) wins a free kick in the defensive half.
Substitution, Genoa. Raffaele Palladino replaces Adel Taarabt.
Foul by Alex Sandro (Juventus).
Goran Pandev (Genoa) wins a free kick on the right wing.
Attempt missed. Gonzalo Higuaín (Juventus) right footed shot from the right side of the box misses to the left. Assisted by Paulo Dybala.
Paulo Dybala scored a hat-trick as Serie A champions Juventus came from 2-0 down to beat Genoa.