Iglesias played for the Piratas de 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.

[6] On June 27, 2014, Iglesias signed with the Cincinnati Reds, to a seven-year contract worth $27 million.

In 2019, Iglesias started the season as the Reds closer and ended the year with career-highs in saves (34) and appearances (68) and a 4.16 ERA.

[16] On June 26, 2022, after the brawl between the Angels and the Seattle Mariners, Iglesias threw a tub of sunflower seeds and a bucket of gum onto the infield.

[17] The Angels traded Iglesias to the Atlanta Braves for Jesse Chavez and Tucker Davidson on August 2, 2022.