Drinčić started his professional career with FK Iskra Danilovgrad in 2016. He established himself as a key starter at the club, making over 160 games and tallying seven goals.[2]
In January 2023, during the winter transfer window, Drinčić was transferred to Belarusian Premier League club Shakhtyor Soligorsk for an undisclosed transfer fee. He suffered an injury that kept him sidelined for a prolonged period. He only made one appearance for the club across all the competition in the Belarusian Cup. With the club, Drinčić won the 2023 Belarusian Super Cup.[4]
Kerala Blasters
On 14 August 2023, Drinčić completed a move to the Indian Super League club Kerala Blasters. He signed a one-year contract with the Blasters and became the first Montenegrin player to sign for the club.[5] He made his debut for the club on 21 September, in the season opener against Bengaluru at home, which the Blasters won 2–1.[6] In Kerala Blasters' 2–1 loss against Mumbai City on 8 October, Drinčić was sent off after pushing Yoell van Nieff from side in the ensuing melée, during the final minutes of the injury time in the second half.[7] He was later handed a three-match suspension.[8] Drinčić scored his first goal for the club in his comeback game against Hyderabad on 25 November, helping the Blasters in a 1–0 win.[9] He would score his second goal of the season on 12 February 2024 against Punjab FC, where he scored the only goal for the team, as the Blasters lost the match 1–2 at full-time.[10]