Murgaš was born in Tajov (Tajó), Kingdom of Hungary, Austrian Empire (now Slovakia). He studied theology in Prešporok (Pressburg, present Bratislava) (1880–82), Esztergom (1882–84), and in Banská Bystrica, where he graduated in 1888. From his youth he was bright, skillful and good at painting and electrotechnology: The vice-head of the school in Esztergom allowed him to use the physics room for experiments, and the Slovak Painters B. Klemens and Dominik Skutecký noticed his talent for painting.

After priestly ordination in 1888, Murgaš worked as a curate. On Skutecký's initiative, Murgaš was accepted at a painting school in Budapest, where he studied from 1889–90. He also studied painting in Munich from 1890-93. He attended both schools while working. He painted sacral pieces and Slovak landscapes and Slovak personalities. It was due to his strong patriotism he exhibited during holidays in the 1890s that he was not allowed to finish his painting studies and had to work as a curate in changing places in the Kingdom of Hungary: in Chrenovec (Nyitratormás), Slovenská Ľupča (Zólyomlipcse), Dubová (Cseres) and in Lopej (Lopér). In Lopej, he painted a large sacral picture of St. George, which is still on the church altar of the village. The central altar painting of St. Elisabeth, in the 14th century Church of St. Elizabeth in the main square of Banská Bystrica, is by Murgaš.

Due to permanent conflicts with the bishop's secretary, Murgaš had to emigrate to the United States in 1896, where he was assigned a Slovak parish in the city of Wilkes-Barre, Pennsylvania. Having no possibility for painting, he started to deal with natural sciences again, especially electrotechnology. He established a laboratory in Wilkes-Barre, in which he primarily investigated radiotelegraphy. His article in the Tovaryšstvo magazine of 1900 shows that his radiotelegraphy studies had achieved a high level. In 1904, he received his first two US patents: the Apparatus for wireless telegraphy and The way of transmitted messages by wireless telegraphy. Further 15 patents followed between 1907 and 1916 (see below). Based on the first two patents, he created the Universal Aether Telegraph Co., which organized a public test of Murgaš's transmitting and receiving facilities in September 1905 (see below). The test was successful, but a storm destroyed the antenna masts three month later, which led to a dissolution of the company.

The tone system is the use of two signals of different frequencies, i.e. Murgaš substituted the "dot" of the Morse code with a higher tone and the "dash" with a lower tone (this is the 1904 patent "The way of transmitted messages by wireless telegraphy").

In Tajov, there is Murgaš's house where he was born, a memorial room, and a symbolic grave with a sepulchral monument of Murgaš at the local cemetery. Jozef Šebo, the founder of the room and monument (now a retired teacher) looks after them very carefully. The memorial room also features originals of pictures, paintings, some unique pieces from his butterfly collection, Models of inventions in wireless telegraphy and documents. One can also see there a minimodel of Murgaš's original antenna masts built by company Universal Aether Telegraph Co. in Wilkes-Barre in 1905.

Murgaš was active in the Slovak expatriates movement, wrote articles for their press, was one of the founders of the Slovak League in America, actively supported the creation of the state of Czechoslovakia, organized a money collection (a fund) of American Slovaks for the creation of Czechoslovakia (1,000,000 USD), and was also a Writer and a signatory of the Pittsburgh Agreement (1918) between Czechs and Slovaks on establishing Czechoslovakia. As a respected personality, he gained trust and support of the highest authorities in the USA for the establishment of Czechoslovakia.

When the United States entered World War I, private radiotelegraphy stations were prohibited, which put an end to Murgaš's pioneer work in this field. After the creation of Czechoslovakia, he returned to Slovakia in 1920, where he taught electrotechnology at a high school, but since he did not find appropriate understanding by the Ministry of Education in Prague, he returned to Wilkes-Barre four months later. He was nominated to be a member of the Federal Radio Commission of the United States in 1925. Murgaš died in Wilkes-Barre four years later.