Cooper, 31, and Esposito, 34, have kept their romance low profile. In January at the Warner Bros. and InStyle Golden Globes party, the pair greeted each other with a kiss on the lips. Asked if there was a love connection to report, Esposito told PEOPLE coyly, "I'm not answering that."
Esposito starred in the 2005-06 WB series Related and this year won, along with her costars, a Screen Actors Guild award for best cast in a motion picture for Crash. She had an eight-month romance with Benjamin Bratt from 1996-97 and later dated All My Children actor Cameron Mathison.
Cooper got his big break in 2001, when he joined the cast of Alias, and in 2005 starred in the short-lived FOX sitcom Kitchen Confidential. He also appeared in last summer's Wedding Crashers and performed opposite Julia Roberts on Broadway in Three Days of Rain.
Asked how he impresses women, Cooper, who loves to cook, told PEOPLE last year, "I like to make stuffed calamari with dried currants, prosciutto, goat cheese and leeks, with a homemade tomato sauce. The problem is, when I cook for a date, I get so maniacal, they see my obsessive side."
He's attracted to independent women, he told PEOPLE: "I love a woman who you can go out to a party with and you don't have to worry. She can take care of herself."
This will be the first marriage for both.
I never even knew they were dating.