Rodríguez's family defected from Cuba on his father's fishing boat when he was eight years old, nearly dying on the journey. Rodríguez played college baseball for the University of Miami before beginning his professional career.

Rodríguez was born in Villa Clara Province, Cuba. His parents, Edilio and Ylya Rodriguez, operated a farm and his father fished.

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. Eddy was unaware of the defection until they were at sea. 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. The family settled in Miami and Rodríguez graduated from Coral Gables Senior High School in Coral Gables, Florida.

Rodríguez received a scholarship from the University of Miami, where he played college baseball as the starting catcher for the Miami Hurricanes baseball team. The Cincinnati Reds drafted Rodríguez in the 20th round of the 2006 Major League Baseball draft. He played in the Reds minor league baseball organization until 2009, reaching Class AA with the Chattanooga Lookouts of the Southern League, when he began playing independent league baseball. Rodríguez played in the American Association of Independent Professional Baseball for the El Paso Diablos in 2009 and the Sioux Falls Pheasants in 2010.

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