Resuming their game which was abandoned after 33 minutes due to the conditions on Tuesday, the two looked set for a goalless draw which would have sent Juve through in the wake of group leaders Real Madrid.
Sneijder changed all that, though, scoring five minutes from the end to bring the Turk Telecom Arena to life and condemn Juve to the Europa League.
It will be a bitter pill to swallow for the Old Lady, whose manager, Antonio Conte, expressed concerns about the pitch as the players went off for half-time, just 14 minutes after 'kicking off'.
That was after Tuesday's game bit the dust with 31 minutes on the clock but, even though snow hung around on Wednesday, the game started for a second time.
Juve, needing just a point to progress, had the first opportunity in the 41st minute when Fernando Llorente fed Carlos Tevez in the box and his right-footed shot went just wide of the far post.
At the other end, Aurelien Chedjou should have done better but struck the ball way over the bar after a good cross from Sneijder.
Paul Pogba was first to test Fernando Muslera at the start of the second half with his central effort from 30 yards kept out by the Galatasaray keeper.
The hosts then almost broke the deadlock in the 57th minute when Burak Yilmaz fed Didier Drogba just outside the area and his goal-bound attempt towards the far post forced Gianluigi Buffon to make a one-handed save.
Gala continued to press forward and Sneijder sent a warning in the 84th minute with a 30-yard attempt that went just wide.
But the Dutchman made no mistake seconds later when he got behind Drogba's headed pass and fired a right-footed shot from the heart of the area towards the far post and out of Buffon's reach.
Stunned, Juve tried to react but they ran out of time.