The Dutchman turned home a precise finish in the final five minutes as Antonio Conte's side were 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 freak snow storms, with play resuming on 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.
Fernando Llorente and Carlos Tevez had both been recalled to a Juve side missing injured forward Mirko Vucinic (knee), while Sneijder made his first start since November 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, Llorente going close after good work from Arturo Vidal, but it was Galatasaray who started stronger when play got back under way in Turkey.
Drogba and Sneijder both probed well for the hosts, without threatening the goal, before Llorente and Tevez 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 who attacked first following the interval as Leonardo Bonucci fired over with an effort from distance early in the second period.
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 defence, 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 goalline intervention from Giorgio Chiellini, with the final whistle prompting joyous scenes a matter of seconds later.
Galatasaray's three points moves them into second behind Real Madrid and into the last 16 draw, with two-time winners Juventus now in the Europa League.