Sneijder made the breakthrough in the 70th minute, with the Dutch international finishing a Galatasaray attack down the left side.
Both teams struggled to create chances in a largely eventless encounter.
Cup success is a consolation for Roberto Mancini's Galatasaray after a disappointing league campaign where the best they can hope for is a second-place finish behind champions Fenerbahce.
Galatasaray, who were also knocked out of the Champions League in the last 16 by Chelsea, will face Fenerbahce for the Turkish Super Cup.
(Writing by Ece Toksabay; editing by Toby Davis)
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