Antonio Conte's team were held in their second straight European game, with Quagliarella's 87th minute goal cancelled out by Bulut's leveler one minute later
Didier Drogba put Galatasaray in front before late goals from Arturo Vidal and Fabio Quagliarella looked to have given Juventus all three points.
Umut Bulut's dramatic leveler, though, ensured that Mancini's tenure began with an encouraging performance and result in Turin.
Antonio Conte made the surprising decision to leave Fernando Llorente on the bench as Stephan Lichtsteiner, Andrea Pirlo and Mirko Vucinic all came into the starting XI.
Mancini was in the dugout against his boyhood club after joining Turkish giant Galatasaray on a three-year deal on Monday. The former Manchester City boss wasted no time in making changes, with six players coming into the first team, including Albert Riera and Felipe Melo.
Mancini went with a five-man midfield after Galatasaray lost 6-1 to Real Madrid in its first Group B match and it was causing plenty of problems for Juventus, which had to use the long ball to try and break through the organised Gala defense.
Drogba gave the visiting side the lead in the 36th minute when a poor back pass from Leonardo Bonucci left Gianluigi Buffon stranded and the former Chelsea striker nipped in to place the ball into the back of an empty net.
On the stroke of halftime, Quagliarella could have scored when he found space at the near post but Fernando Muslera was alert to thwart the danger.
Juventus continued to struggle in the second half and Conte introduced Llorente as he looked to find a way back into the game.
Conte's men got their reward in the 78th minute when Quagliarella was brought down by Nordin Amrabat and Vidal coolly stepped up before smashing the resulting penalty into the top corner to give Muslera no chance.
Galatasaray lost the momentum and the home side capitalized when Quagliarella got on the end of a perfectly-weighted Pirlo cross to head home in the 87th minute.
With the home crowd still celebrating Quagliarella's goal, Galatasaray struck back immediately and got a goal from the kickoff.
Drogba's flick was met by Bulut who got enough power on his shot to guide it past Buffon in the Juventus goal.
Carlos Tevez and Llorente went close late on for the Serie A champion, but they to deny Mancini a dream start.