In 2005, Iglesias won the national (senior) flyweight title as a 16-year-old when Olympic champion Yuriorkis Gamboa was out injured.

In 2006, he competed in January at the national (senior) championships at bantamweight and lost in the semifinal to Idel Torriente, in September he had put on 6 more kilograms and won the 2006 world title at lightweight in Agadir, Morocco.

[3] In 2008 he won the junior welter national senior championship, he beat Richard Poll 12:9 in the final and was sent to the Olympic qualifier in Trinidad where he beat another teenager Javier Molina in the semifinal to qualify for the 2008 Summer Olympics where he lost his semi to defending champion Manus Boonjumnong.

[4][5] In the 2012 Olympic Games in London, Iglesias won the gold medal by defeating Denys Berinchyk from Ukraine by a score of 22–15.

[7] In the 2020 Olympic Games in Tokyo, Iglesias won a gold medal by defeating Pat McCormack from Great Britain by a score of 5–0.