Initially the club's second-choice goalkeeper, he made his league debut on 31 August 2012, playing the full match in a 1–0 defeat at home to HNK Cibalia.

On 30 July 2019, in a Champions League qualifying 3–0 victory over Saburtalo Tbilisi, Livaković broke Dražen Ladić's record for most minutes from the start of Dinamo's season without conceding a goal (413), set in 1995.

Livaković saved Cătălin Golofca's attempt which would turn out crucial as Dinamo won the shootout 6–5 and progressed to the third qualifying round.

[23] On 26 September 2024, he saved his compatriot Franjo Ivanović's 90+2 minutes penalty kick against Union Saint-Gilloise in a 2-1 win, UEFA Europa League home match.

[26] In May 2018 Livaković was named to Croatia's preliminary 32-man squad for the 2018 FIFA World Cup in Russia, and was eventually part of the team that finished runners-up after losing the final against France.

[27] He made his competitive debut against England in the 2018–19 Nations League match held on 12 October 2018 at Stadion Rujevica in Rijeka, which ended in a goalless draw.

His father Zdravko Livaković is a construction engineer and a former State Secretary in the Ministry of the Sea, Transport and Infrastructure during the mandate of Božidar Kalmeta.