Šušak was born in Široki Brijeg, then Independent State of Croatia, today Bosnia and Herzegovina. His father and oldest brother were fighting in Ustaše Army and went missing in May 1945, presumably killed by Partisans after the Nazis had retreated.
After high school graduation, Šušak studied mathematics in Rijeka. When he received a draft call to the Yugoslav People's Army, Šušak decided to emigrate. Allegedly with the help of Croatian Franciscan priests, he illegally crossed the Yugoslav-Austrian border. This claim is disputed by some, particularly Paul Hockenos, who in his book, Homeland Calling, states Šušak probably left Yugoslavia, like most Croatians, as a guest worker.
From Austria, Šušak went to Canada. There he was doing all kinds of odd jobs for living. His political opponents in 1990s mockingly called him "Pizza man", since he worked in a pizzeria for some time.
In 1973, he married another Croatian immigrant, Đurđa a social worker at the time. They had two daughters and a son named Tomislav, and lived peacefully in suburbia. However, Šušak was at the same time one of the most active Croatian political immigrants in Canada. He organised Croatian schools, football clubs etc.
In 1979 Šušak with his immigrant friends, planned to place a pig in a coffin in front of the Yugoslav Embassy, with either the intention of letting it loose on embassy grounds or possibly killing it. However, this plan was disrupted by the Ontario Humane Society, who rescued the animal.
During his exile, Šušak was associated with Croatian Franciscans in Canada, especially with their mission in Norval, which was politically active. Šušak and Norval priests were hosts to Partizan General turned dissident Franjo Tuđman, during his first visit to Croatian immigrants in Canada in late 1980s. During that visit, Tuđman and Šušak became friends and built a bond that would last until Šušak's death.
Šušak and his circle managed to raise huge amounts of money among Croats in North America that helped Tuđman finance his rise to power in Croatia. Šušak went back to Croatia in 1989, became high official in Tuđman's party, Croatian Democratic Union (HDZ) and after the party's win in the 1990 election (first free election in Croatia and rest of Yugoslavia since World War II), Minister of Immigration.
In 1991 Croatia and Slovenia declared independence from Yugoslavia. In September 1991, Šušak was named minister of defense while Croatia was at war with rebel Serbs supported by the Yugoslav People's Army, who managed to occupy a quarter of Croatia's territory. When war spread to Bosnia and Herzegovina, a multi-ethnic former Yugoslav republic, Šušak led a group of Croatian officials who believed Croatia should annex parts of BiH where Croats were a majority, including western Herzegovina, a hotbed of Croatian nationalism and Šušak's home region.
That led to the creation of a Croatian state "Herceg-Bosna", and war between Bosnian-Herzegovinian Croats and Bosniaks, who had previously allied against Serb aggressors. In 1994, a group of HDZ officials including current Croatian President Stjepan Mesić left the party because of Tuđman and Šušak's politics in Bosnia and Herzegovina.
In the same year, after USA-led diplomatic effort, Croats and Bosniaks reconciled, which led to a numerous offensives against Serbs in Croatia and Bosnia in 1995. After Operation Flash and Operation Storm against rebel Croatian Serbs, Bosnian Serbs suffered a series of defeats as well, so they were forced to start peace negotiations that produced the Dayton Agreement, where Šušak was one of the key Croatian negotiators in Dayton.
After the war, Šušak stayed in politics and became unpopular abroad in the international community, though he cultivated some notable friendships too, especially with U.S. Secretary of Defense William Perry. Perry represented Washington at Šušak's funeral but also said he came as a personal friend. In his eulogy, he paraphrased Shakespeare's verse saying "Now there goes a man, we shall never look upon his like again". In Croatia, Šušak was considered a key figure in the successful war effort by some, and a ringleader of high-ranking corrupted officials by others.
A heavy smoker, Šušak was diagnosed with throat cancer. He was treated in Walter Reed Army Medical Center, just like Tuđman, who was also diagnosed with cancer at about the same time. Gojko Šušak died in Zagreb at the age of 53.