After deciding to become a priest and joining the seminary, he attended a high school in Dubrovnik and then the College of Theology of Vrhbosna in Sarajevo beginning in 1987.

In 2022, Kutleša succeeded Marin Barišić as the archbishop of Split-Makarska, became a member of the Dicastery for Bishops, and was elected as the president of the Episcopal Conference of Croatia.

Dražen Kutleša was born in Duvno, present-day Tomislavgrad, in Bosnia and Herzegovina, to father Krešo and mother Danica née Ćurić, both from the village of Prisoje.

Because his father was, like many men from his region, a gastarbeiter (guest worker) in Austria and West Germany, Dražen and his brother Grgo were raised by their mother.

He chose Psalm 86:12, "I will praise thee, O Lord my God, with all my heart: and I will glorify thy name forevermore", as his priestly motto and celebrated his first mass in Prisoje on 25 July 1993.

In 1995, he was sent to study canon law at the Pontifical Urban University in Rome with a stipend from the Congregation for the Evangelization of Peoples, living on the premises of the Society of St. Peter the Apostle.

That same year, Kutleša was appointed vice chancellor of the diocese, and in 2000, he became personal secretary to Perić, who succeeded Žanić as the bishop of Mostar-Duvno.

On 18 June 2001, Kutleša received his PhD after defending his dissertation in front of Vittorio Pio Pinto, a judge of the Roman Rota, the highest court of the Catholic Church.

From February 2011, Kutleša was also an associate of the Congregation for Divine Worship and the Discipline of the Sacraments for cases of granting exemption from difficult and unconsummated marriage.

[2] On 17 October 2011, Pope Benedict XVI named Kutleša bishop coadjutor of Poreč and Pula with special powers in Croatia.

[5] Kutleša's appointment occurred during a dispute over the ownership of the Dajla monastery and its accessory properties between the Italian Benedictines and the Diocese of Poreč and Pula.

[6][9] However, just when the commission's decision was supposed to take effect, Pope Francis was elected on 13 March 2013, and Secretary of State Cardinal Tarcisio Bertone, who supported the Benedictines, was replaced on 15 October 2013.

[11] In August 2017, the government of Andrej Plenković donated the state-owned property Villa Idola in Pula to the Diocese to house its offices.

The minister of state assets, Goran Marić, said that his Ministry could not restore the property confiscated from the Church by the communist authorities and that this donation "rights the wrongs from the past".

[12] On 11 June 2020, Pope Francis appointed Kutleša archbishop coadjutor of Split-Makarska,[13] naming him apostolic administrator of Poreč and Pula as well.