Koschat was born on 8 August 1845 in the Viktring district of Klagenfurt. From 1865 to 1867 he attended the Technical University of Vienna, studying chemistry, but he did not obtain a degree. Instead his imagination was captivated by music, and he formed his first vocal quartet in 1866. He joined the Vienna State Opera in 1867, singing bass. In 1874 he joined the choir at St. Stephen's Cathedral. From 1877 until 1906 he performed with the Koschart Quintet. That year he also joined the choir at the Hofmusikkapelle and served as soloist there. As a composer he was known for his Kärntnerlieder, folk songs of Carinthia arranged by himself, or else his own compositions set in that style. With his men's quartet and quintet he toured Europe and the Americas, where the performances of his Carinthian music were enthusiastically received. For most of his life he resided in Vienna. Nonetheless he was made an honorary citizen of Klagenfurt in 1907 owing to the local nature of his work. Koschat died in Vienna, at his home on Strobachgasse in the 5th district, on 19 May 1914. He was buried at
Annabichl near Klagenfurt.