Iglesias played for Isla de la Juventud in the Cuban National Series , and played for the Cuban national baseball team , appearing in the 2013 World Baseball Classic and 2013 World Port Tournament . 
In September 2013, Iglesias attempted to defect from Cuba. He hid in the mountains of Isla de la Juventud , but was caught and was detained.   In November of that same year, Iglesias successfully defected from Cuba.  He established his residency in Haiti , held an open tryout in Mexico in December 2013,  and signed with the Cincinnati Reds on June 27, 2014, to a seven-year contract worth $27 million.  The Reds formally added him to their 40-man roster when he arrived in the United States on August 12. 
In spring training in 2015, the Reds evaluated Iglesias as a starting pitcher .  He started the season in the Reds' Opening Day starting rotation , and made his major league debut on April 12.  He had a couple of stints on the disabled list, limiting him to just 18 games, 16 of them starts. His record was 3–7 with 104 K's. The following season, after beginning the season in the rotation, Iglesias was moved to the bullpen. Iglesias served as setup man and closer for the Reds, finishing with 6 saves in 37 games.