Juve found themselves 2-0 behind after just seven minutes, but Dybala's double pulled them level and the Argentine put the gloss on things with an excellent third that took his Serie A tally to 50 following Juan Cuadrado's effort.
An own goal from Miralem Pjanic just 18 seconds into his 50th appearance for the club got Juve off to a terrible start and matters were made worse when Andrej Galabinov scored a penalty after referee Luca Banti used VAR to look at Daniele Rugani's challenge on the Bulgarian.
It was the second time in as many matches Massimiliano Allegri's side found themselves on the wrong side of a spot-kick decision based on video technology, but the system worked in their favour after Dybala had got them going with an instinctive finish.
Banti deemed Darko Lazovic to have handled a shot from Mario Mandzukic after watching a replay and Dybala restored parity from 12 yards deep in first-half stoppage time.
Cuadrado beat Mattia Perin to put the visitors ahead for the first time and Dybala fired a sweet near-post strike beyond the goalkeeper to ensure Juve escaped with a second win from as many Serie A matches.
An electric start saw Genoa take a surprise lead in the opening minute, Goran Pandev's cross from the right ending up in the back of the net after Giorgio Chiellini deflected it onto Pjanic.
Perin produced a sensational save to stop Santiago Gentiletti heading into the back of his own net and Genoa had a chance to move two up when referee Banti used VAR to award them a penalty for Rugani's tackle from behind on Galabinov.
Stephan Lichtsteiner was booked for his protests, with the Genoa forward having appeared to be offside from Adel Taarabt's touch, but Galabinov was unforgiving and fired his spot-kick down the middle.
Juve, without injured midfielder Claudio Marchisio, did not dwell on the setback and halved the deficit when Dybala met Pjanic's cutback with a cool, first-time finish beyond Perin in the 14th minute.
As Genoa looked to threaten on the break they came under pressure and Perin produced a remarkable double save to keep out Dybala and Mandzukic in quick succession.
Juve were not to be denied an equaliser before the break, though. Banti used the VAR system again to penalise Lazovic for blocking a shot from Mandzukic with his arm and Dybala drilled the resultant spot-kick into the top-right corner.
Allegri's side struggled to create and, after Blaise Matuidi replaced Sami Khedira to offer some additional dynamism in midfield, Juve breached the Genoa defence again in the 62nd minute when Cuadrado beat Diego Laxalt and picked out the top-left corner.
Rodrigo Bentancur made his Juve debut as a replacement for Pjanic before Gianluigi Buffon – named UEFA Goalkeeper of the Year in midweek – kept out Raffaele Palladino as Genoa pushed for an equaliser.
Dybala made the scoreline more flattering by arrowing a low shot in at the near post in the 92nd minute, his landmark goal coming in fine fashion.
