Bishop Šaško was born in the small village of Ðivan, near Vrbovec, on 1 August 1966. He studied theology and philosophy at the Catholic Theological Faculty of the University of Zagreb, at the Pontifical Atheneum of St. Anselm, and at the Pontifical Gregorian University.
In 1991, he was ordained a deacon for the Archdiocese of Zagreb. On 28 June 1992, Šaško was ordained a priest for the Archdiocese of Zagreb. In 1996, Mgr. Šaško started working as an assistant in the Department of the liturgy of the Zagreb Catholic Theological Faculty. From the year 2000, he worked as a docent in the same department. From 2002 to 2007, he was head of the Department of liturgy.
From 2001, he is a member of the Assembly of prebendaries of the Zagreb cathedral. In 2008 he was appointed canon of the Cathedral Kaptol of Zagreb. In January 2005, he was appointed President of the Commission for the establishment of the Catholic University of Croatia, and still today, after the University opening in 2006, continues as the President of the Commission of the University.
In April 2005, he was elected associate professor in the Department of the liturgy of Zagreb Catholic Theological Faculty. He is a member of the editorial board of the scientific journal Bogoslovska Smotra.
On 11 February 2008, Pope Benedict XVI appointed Mgr. Šaško Auxiliary Bishop of Zagreb and Titular Bishop of Rotary. Cardinal Josip Bozanić ordained him on 28 March 2008.
Mgr. Šaško is Bishop Delegate to COMECE since November 2022.