Kravtsiv was taught how to play chess on his sixth birthday by his father. Later, he trained in a Lviv Chess Club. In the 2008 World Mind Sports Games in Beijing, Kravtsiv won the gold medal in the men's individual blitz event. In 2010, he tied for 1st-6th places with Dmitry Kokarev, Alexey Dreev, Maxim Turov, Baskaran Adhiban and Aleksej Aleksandrov in the 2nd Orissa Open tournament in Bhubaneshwar. In 2011, Kravtsiv won the 3rd Chennai Open. The following year, he tied for 1st–5th with Pentala Harikrishna, Parimarjan Negi, Tornike Sanikidze and Tigran Gharamian in the Cappelle-la-Grande Open. He tied for first with Andrei Volokitin and Zahar Efimenko in the 2015 Ukrainian Chess Championship, held in his native city of Lviv, finishing second on tiebreak. In 2016, Kravtsiv won the Riga Technical University Open edging out Hrant Melkumyan, Arturs Neiksans, Aleksey Goganov and Jiri Stocek on tiebreak score.