Balogh attended the Budapest Conservatory from 1905 to 1917. His teachers included Béla Bartók for piano and Zoltán Kodály for composition, the two subjects in which he won the Franz Liszt Prize. Balogh became close friends with both men; indeed, in 1927 he arranged for Bartok to make his first concert tour in the United States.
After completing his course at the Budapest Conservatory and further piano studies with Leonid Kreutzer at the Berlin Conservatory, Balogh moved to the United States in 1924. Settling in New York, he established a successful career as both soloist and accompanist; in the latter capacity, he played with celebrated musicians including Fritz Kreisler, Lotte Lehmann, and Grace Moore.
In 1936, Balogh married Malvina Schweizer, a university professor. They continued to live in New York until 1960, when they moved to Washington, DC pursuant to Balogh's acceptance of a position teaching at the Peabody Conservatory in Baltimore, Maryland.