Growing up, Bazardo played in Venezuelan youth leagues, where one of his competitors was future teammate (on two different MLB clubs) Miguel Cabrera. Bazardo got his break in 2000, when the Florida Marlins signed him as an undrafted free agent. His first season in the organization was in 2001, when he played for the now-defunct VSL Marlins in the Venezuelan Summer League, going 7–2 with a 2.43 earned run average (ERA). The Marlins moved him up to the single-A Jamestown Jammers in 2002, where he posted a 5–0 record with a 2.72 ERA and six saves. In 2003, he was promoted to the single-A Greensboro Bats and used primarily as a starting pitcher. Although he had a mixed record (9–8) his ERA remained a low 3.12. Typical was a game in August against the Hagerstown Suns in which he allowed just two runs over eight innings in a 2–0 loss.