Šarić's chess career began at the age of seven. He quickly demonstrated his extraordinary talent by winning numerous youth competitions. In 2001 and 2002 he won the European U12 and U14 championships. However, Šarić's most significant victory came in 2007 when he won the U18 World Youth Chess Championship in Antalya.
In addition to the aforementioned achievements, Šarić won the Heart of Finland tournament in 2010, Reykjavik Open in 2014, and the 19th HIT Open in Nova Gorica in 2014. He also had an extraordinary performance in the 2014 Chess Olympiad in Tromsø, with 8 out of 10 points.