Mrsic was born in the Rasavci village near Prijedor, SR Bosnia and Herzegovina, Yugoslavia, to father Momir and mother Radojka. His father got a job in Sweden in the late 1960s. Dragomir moved with his mother and two older siblings to Sweden aged a month-and-a-half, and was brought up in Fittja, a suburb of southern Stockholm. His father worked as a chef at Grand Hôtel in the city, his mother as a cleaner at the local school. His father was a functioning alcoholic. At 14 years of age he witnessed his brother's friend's suicide jump. His two siblings, sister Ranka and brother Radomir "Rajko", died of drugs use.

He has trained Taekwondo since his youth, and has a black belt. His childhood idol was Bruce Lee. At 18 years of age, in 1987, he became Nordic Champion in Taekwondo. In 1990, he and friend Liam Norberg were involved in the robbery of bank Götabanken; he stood guard some hundred meters away. He was arrested and sentenced to three years and six months in prison for the robbery, and served his sentence at Hall Prison. He studied Buddhism in prison. After serving his sentence, he entered a Sports college. He then started a gym, and became a Sports consultant. He was a top program trainer in the Swedish Olympics Committee in 2000–04.

After a few minor roles, such as in the Wallander film Kuriren (2009), he got his big break out in 2010 as Mrado, in the critically acclaimed film Easy Money (2010).

In 2014, he co-starred in the Tom Cruise film, Edge of Tomorrow.

In January 2015, he gave his voice and likeness to a character called Dragan for the game Payday 2. Ironically, he is the only person who has given their likeness to a character in the crime-themed game who has actually been involved in illegal activities.