[1][2] The Rodríguez family, consisting of Eddy, his parents, his sister Yanisbet, and his cousin Carlos, defected from Cuba when he was eight years old on his father's fishing boat.

[1][2][3] Their boat nearly capsized in a storm and they ran out of food, resorting to eating ground coffee beans, before they were discovered by the United States Coast Guard.

[6] Rodríguez played in the American Association of Independent Professional Baseball for the El Paso Diablos in 2009 and the Sioux Falls Pheasants in 2010.

[6] In 2011, Rodríguez returned to organized baseball when he signed a minor league contract with the San Diego Padres.

[6] He was promoted to the Padres on August 1, 2012, from the Lake Elsinore Storm of the High–A California League, after starting catcher Yasmani Grandal was placed on the disabled list.

[13] In 57 games split between the Double–A Trenton Thunder and Triple–A Scranton/Wilkes-Barre RailRiders, he compiled a .171/.202/.261 batting line with three home runs and 20 RBI.

[16] The Twins released him after spring training, and he signed a minor league contract with the Yankees on April 3, 2017.