Sturm's 14th title defence proved his last as his Australian opponent finished strongly to win a 112-116 116-112 116-112 decision.
It was IBF champ Geale's 28th victory in 29 contests and a first defeat since 2006 for Sturm.
"I have to accept the decision, that is boxing," said the German. "He did well on his side, he boxed awesomely and gave his best and I simply have to accept it. It's a bitter defeat, but I had my chances."