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Carter dominated the evening session in the German capital to take the 60,000-euro top prize and add to the titles he won at the 2009 Welsh Open and 2010 Shanghai Masters.
Hong Kong's 35-year-old Fu looked the stronger player in the early stages but struggled with his safety and potting at the business end.
Fu had been seeking his second ranking title having won the 2007 Masters.
"I didn't have many chances tonight. Every time I came to the table I was faced with a tough shot," said Fu.
"Ali played very well so all credit to him. It was nice to be in a final again as the crowd was great - hopefully I can come back next year and win it."