The Dutchman converted a precise finish in the final five minutes as Antonio Conte's side was sent crashing out of the Champions League.
The Dutchman latched on to Didier Drogba's flick to fire low past Gianluigi Buffon in the closing minutes as Roberto Mancini's side battled past the Bianconeri in treacherous conditions in Istanbul. The fixture was halted after 31 minutes on Tuesday due to snow and hail, with play resuming Wednesday afternoon.
On a heavy, muddy pitch, both sides were largely limited to direct passages of play when the game restarted, with Juventus coach Antonio Conte appearing very unhappy with the match officials at the break, presumably due to the condition of the playing surface.
He would doubtlessly have been even more angry when Sneijder, a former Inter player, fired home in the 85th minute. The goal secured Galatasaray the three points it needed to move second, sending two-time winner Juventus into the Europa League.
Fernando Llorente and Carlos Tevez had been recalled to a Juve side missing injured forward Mirko Vucinic (knee), while Sneijder made his first start since Nov. 1 for Galatasaray after shaking off a recurring thigh problem.
The visitors had carved out the only clear-cut chance prior to Tuesday's suspension in play, with Llorente going close after good work from Arturo Vidal, but Galatasaray started stronger when play got back underway in Turkey.
Drogba and Sneijder probed well for the hosts without threatening the goal. Llorente and Tevez then linked impressively, only for the former to lose his footing as both sides struggled with the conditions.
Referee Pedro Proenca drew the half to a close after 14 minutes of play, and it was Juve that attacked first following the interval as Leonardo Bonucci fired over with an effort from distance early in the second half.
Drogba did well to shield the ball in an advanced position at the other end, allowing Sneijder to fire wide, with the Ivorian then forcing a smart right-handed save from Gianluigi Buffon following a vicous effort.
Claudio Marchisio saw an attempt on goal saved after a rare flowing move down the left in the 79th minute before Juve's defense, which boasts seven consecutive clean sheets in serie A, was finally breached with five minutes left. Drogba headed a long ball into Sneijder's path, with the Netherlands international cooly firing past Buffon.
The hosts almost made it 2-0 late on but Burak Yilmaz was denied by a goal-line intervention from Giorgio Chiellini, with the final whistle prompting joyous scenes seconds later.