The trademark lawsuit filed in Manhattan federal court also calls on Costco to forfeit profits from the ring sales, plus damages of $2 million per infringement.
Tiffany said it believes hundreds, if not thousands, of Costco members bought engagement rings they believed were authentic Tiffany products, but in fact were not.
Luxury retailers often sue to stop sales of alleged imitation products they believe cut into their own revenue streams and customer goodwill.
"This is not the kind of behavior people expect from a company like Costco, and this case will shed a much needed light on this outrageous behavior," said Jeffrey Mitchell, a lawyer for Tiffany.
Mitchell also noted in the statement that Tiffany maintains an "aggressive and rigorous intellectual property program, and regularly takes actions against counterfeiters and infringers."
A spokesman for Costco was not immediately available.
In November 2012, a person who was shopping at a Costco in Huntington Beach, Calif., complained to Tiffany that she was disappointed to see Costco offering for sale what were promoted on in-store signs as Tiffany diamond engagement rings, according to the lawsuit.
A subsequent investigation revealed rings in a display case at the Huntington Beach Costco labeled with the Tiffany brand and that a salesperson there referred to them as Tiffany rings when they were not, according to the lawsuit. Tiffany also discovered that Costco was not using the Tiffany trademarks online for the same products at the same time to avoid detection, according to the lawsuit.
The case is Tiffany & Company and Tiffany (NJ) LLC v. Costco Wholesale Corp, U.S. District Court, Southern District of New York, 13-1041