Ciro Immobile gave Genoa the lead against the club which still owns half his contract, and the home side had several chances to double its advantage as Juventus seemed headed for its first league defeat since May, 2011.
However, Mirko Vucinic and Kwadwo Asamoah came on in the second half to completely change the game. Emanuele Giaccherini levelled before Vucinic gave Juventus the lead from the penalty spot.
Asamoah sealed the result eight minutes from time.
Napoli, Lazio and Sampdoria also won to maintain their 100 per cent records after three matches. Roma gave up a two-goal advantage to lose 3-2 to Bologna, which picked up its first points of the season.
Juventus appeared to have one eye on Wednesday's crunch Champions League opener with Chelsea and rotated its squad.
"We suffered in the first half and we knew how difficult it would be to play in such a stadium," Juventus stand-in coach Massimo Carrera said. "Genoa started well and made us suffer quite a bit. Then we managed to up the pace and got back into the game.
"Genoa's long balls put us in difficulty also because some players aren't at their best form yet. We have to improve in several areas, there were too many counterattacks today. We'll think about Chelsea from tomorrow, our heads were focused on Genoa, because we knew it would be a difficult test."
The Bianconeri struggled from the outset and almost found themselves behind early on when first Gianluigi Buffon hastily palmed clear Bosko Jankovic's free kick, before Immobile missed a great opportunity.
The Italy under-21 international raced clear one on one with Buffon but rushed his effort and fired wide of the left upright.
Immobile made up for his earlier errors by giving Genoa the lead with a scuffed shot through a sea of bodies after some static defending from Juventus.
Genoa could have doubled its lead 10 minutes from halftime when former Juventus striker Marco Borriello dribbled past Giorgio Chiellini, but Leonardo Bonucci slid in to block his effort.
Soon after Alessandro Matri should have levelled after beating the offside trap but he blazed over from eight yards.
Juventus had another great chance five minutes into the second half when Sebastian Giovinco's effort crashed off the right upright and straight back to Giaccherini who spooned his shot over from the middle of the six-yard box.
Genoa should have extended its lead when Jankovic picked out Borriello all alone in the area, but he headed onto the crossbar. Buffon then pulled off a stunning save to deny Andrea Bertolacci.
Straight from that chance, Juventus equalized when Giaccherini fired in from outside the area following a good assist from Vucinic, who had just come off the bench.
Juve took the lead 12 minutes from time when Mario Sampirisi fouled Asamoah in the area and Vucinic stepped up to fire the penalty into the bottom right corner.
It sealed the result shortly after. Vucinic's cross was deflected by Andreas Granqvist straight into the path of Asamoah, who bundled it in.
In Naples, Napoli got off to the perfect start when Parma goalkeeper Antonio Mirante brought down Goran Pandev in the third minute. Edinson Cavani stepped up to the spot and fired his penalty into the right hand corner, past Mirante's outstretched hand.
Pandev got on the scoresheet in the 39th minute after Marek Hamsik raced clear and unselfishly slotted the ball across goal for the Macedonia striker to stroke home.
Marco Parolo got Parma back into the game on the stroke of halftime, shortly after Napoli midfielder Blerim Dzemaili had hit the right hand post with a free kick.
Lorenzo Insigne, who had a fantastic debut for Italy midweek, scored his first Serie A goal to seal Napoli's 3-1 win.
In the Serie A lunchtime fixture, Lazio also extended its perfect record with a 3-1 win over Chievo Verona. Brazil playmaker Hernanes scored twice and Germany striker Miroslav Klose also found the target for Lazio. Both players have three goals in three league matches.
Sergio Pellissier scored for Chievo with a penalty in the 83rd.
Newly-promototed Sampdoria also has three wins from three in Serie A after beating Pescara 3-2. It remains just behind the top three after starting the season with a point deduction for its role in the match-fixing scandal.
Maxi Lopez scored twice and Marcelo Estigarribia added a third. Mervan Celik and Gianluca Caprari scored Pescara's first Serie A goals for 20 years.
Roma thought it was cruising to victory after establishing a 2-0 lead. Francesco Totti hit the crossbar with a great effort which rebounded to Alessandro Florenzi, who headed in. Erik Lamela doubled its lead in the 16th minute.
However, Bologna was back on level terms in the second half with two goals in a minute from Alberto Gilardino and Alessandro Diamanti.
Gilardino doubled his tally in stoppage time to hand Bologna all three points.
Udinese also let slip a two-goal lead but picked up its first point of the season with a 2-2 draw at Siena.
Dusan Basta and Antonio Di Natale scored inside six minutes but the game changed when Andrea Lazzari was sent off for Udinese after picking up his second yellow card seven minutes into the second half.
Emanuele Calaio pulled one back for Siena before Ze Eduardo levelled from the penalty spot.
Elsewhere, Luca Toni scored just a minute into his Fiorentina comeback to help it beat Catania 2-0. Stevan Jovetic nabbed the opener with his fourth goal in three games.