He died on Thursday evening at the age of 86, as announced by the archbishopric.
Kothgasser headed the archdiocese from 2003 to the end of 2013 and had previously led the diocese of Innsbruck since 1997.
Cardinal Christoph Schönborn honoured the deceased as a "model of gentleness" and as "cheerful, cordial and close to people".
"We in the Bishops' Conference have gratefully learnt to appreciate his gift of mediation and reconciliation," Schönborn told the Viennese press agency Kathpress. Kothgasser was "a good shepherd in the best sense of the word", said the Archbishop of Vienna and long-standing Chairman of the Austrian Bishops' Conference.
Born on 29 May 1937 in Sankt Stefan im Rosental in Styria, Kothgasser joined the Salesian Order at the age of 18. He was ordained a priest in 1964.
Among other things, he studied at the Pontifical Salesian University in Rome, where he later also taught. In October 1997, Kothgasser was appointed Bishop of Innsbruck by Pope John Paul II; five years later, the Salzburg Cathedral Chapter elected him as the new Archbishop.
Title of a "Primas Germaniae"
Kothgasser was also known for his long-standing commitment to the Catholic Church's relations with the Eastern Churches.
The Vienna-based Pro Oriente 2015 foundation made him an honorary member in 2015.
Until 2017, Kothgasser was Grand Prior of the Austrian Lieutenancy of the Knights of the Holy Sepulchre of Jerusalem.
The archbishops of Salzburg still hold the historic title of "Primas Germaniae".
This is no longer associated with a canonical, but rather a liturgical primacy of honour.
The title was initially held by the Bishop of Magdeburg.
After the Peace of Westphalia (1648), when Magdeburg became Protestant, it was transferred to Salzburg.
After his successor Franz Lackner took office, Kothgasser moved to the neighbouring diocese of Innsbruck; he moved into a convent of sisters in Baumkirchen in Tyrol.
A new bishop should "be free and not have the former bishop behind him", said the former bishop at the time.
He last lived in the Salzburg seminary from 2022.