Zubčić took part in FIS races for the first time in January 2009 as a 15-year-old . A month later he was a participant in the European Youth Olympic Winter Festival in Szczyrk , where an eleventh place was his best result. In March 2010 the debut in the European Cup followed . A specialization in giant slalom and slalom became apparent early on, but he also competed in some super-G races. On October 28, 2012, Zubčić made his debut in the World Cup and reached 55th place in the giant slalom in Sölden . A long period of unsuccessfulness ended in January 2013 with the first victory in an FIS race. He was also able to establish himself in the European Cup: only a little later, on January 27th, he achieved his best result at this level with fourth place in the slalom on the Großer Arber .
In 2011 and 2013, Zubčić took part in world championships, followed by participation in the Winter Olympics in Sochi in February 2014 . There he was eliminated in the first run of the giant slalom. At the end of the 2013/14 season he won two Croatian championship titles. In his tenth World Cup appearance on December 12, 2014, he finished 25th in the giant slalom in Åre , winning his first World Cup points. He had a surprise nine days later on the Gran Risa in Alta Badia , where he also advanced to seventh place in the giant slalom with the high start number 64. He scored his first slalom World Cup points on January 6, 2015 at his home race on Slijeme near Zagreb .
Zubčić only partially confirmed these results in the 2015/16 World Cup . Three top 15 placements and the first points in a combination competition were offset by numerous failures. In the World Cup 2016/17 he did not get past 14th place, in the World Cup 2017/18 two eighth places were his best results. At the 2018 Winter Olympics , he finished 12th in the combined and 24th place in the giant slalom. He achieved another eighth place during the 2018/19 World Cup
Zubčić made the final breakthrough to the top of the world in the 2019/20 season in the giant slalom discipline. After finishing eighth again on December 8, 2019 in Beaver Creek , he narrowly escaped a serious accident two weeks later in Alta Badia: As soon as he crossed the finish line, he was able to avoid a collision with an untrustworthy marshal by just a few centimeters; Despite this incident, it was enough for him to finish 10. The first World Cup podium place was achieved by Zubčić on January 11, 2020 with second place at the Chuenisbärgli in Adelboden , behind the Slovenian Žan Kranjec . It was the first Croatian podium since Ivica Kostelić five years earlier. On February 22, 2020, Zubčić celebrated his first World Cup victory in Naeba , with a clear best time in the second run of the giant slalom from twelfth place to the top. He achieved the first victory of a Croat in this discipline. Ten days later, another second place followed in Hinterstoder .