Juve boss Antonio Conte picked an attacking lineup featuring Carlos Tevez and Mirko Vucinic but they were restricted to keeping the ball deep early on, with Gala happy to sit and wait for chances to counter.
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The Old Lady suffered a setback midway through the first half as Vucinic was forced off after appearing to pull his hamstring, with Fabio Quagliarella introduced in his place.
Gala grew in confidence as the first half progressed and took the lead 10 minutes before the interval, with Drogba applying the finishing touch.
The veteran seized on a sloppy backpass by Leonardo Bonucci, controlling before tucking into an empty net to put Mancini's side in front.
Conte then threw caution to the wind, replacing a defender with a striker as Fernando Llorente came on for Bonucci with 20 minutes to go.
Llorente's physical presence soon had an impact as the Spaniard headed over from six yards, before the hosts won a penalty 11 minutes from full-time.
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Quaglirella was the man who earned the spot kick after forcing a foul from Nordin Amrabat, with Vidal slamming into the top corner to restore parity.
Juve threw everything forward in the closing stages and looked to have snatched all three points late on, with Quagliarella heading into the top corner from close range.
There was still time for further late drama as Gala went straight up the other end to draw level, as Umut Bulut caught the visitors cold to turn home Drogba's flick and leave Juve ruing defensive errors.