Carew is the son of Olga Teoma, and Eric Carew, Sr. a painter. Carew is a Zonian and was born to a Panamanian mother on a train in the town of Gatún, which, at that time, was in the Panama Canal Zone. The train was racially segregated; white passengers were given the better forward cars, while non-whites, like Carew's mother, were forced to ride in the rearward cars. Traveling on the train was Dr. Rodney Cline, who delivered the baby. In appreciation for this, Mrs. Carew named the boy Rodney Cline Carew.
At age 14, Carew and his family immigrated to the United States and settled in the Washington Heights section of the borough of Manhattan, New York City. Although Carew attended George Washington High School, he never played baseball for the high school team. Instead, Carew played semi-pro baseball for the Bronx Cavaliers, which is where he was discovered by Minnesota Twins' scout Monroe Katz (whose son, Steve, played with Carew on the Cavaliers). Katz then recommended Carew to another Twins' scout, Herb Stein, who along with Katz signed Carew to an amateur free agent contract (at the Stella D'Oro Restaurant in the Bronx) on June 24, 1964.
Starting his minor league career, Carew was assigned to play second base with the Melbourne (FL) Twins in the Cocoa Rookie League and hit .325 over the final 37 games of the season. Promoted to the Single-A Orlando Twins in the Florida State League the following season, Carew hit .303 with 20 doubles, eight triples and one home run. Carew spent the 1966 season back at Single-A, but this time with the Wilson Tobs in the Carolina League.