He relocated to NYC in 1930, travelling on the same ship as Don Azpiazú’s Havana Casino Orchestra with singer Antonio Machín (1900-1977), who are widely credited for introducing Cuban music to the USA in 1931 with their crossover hit "El Manisero" (The Peanut Vendor). Bauzá learned trumpet in two to three weeks to record with Machín’s group Cuarteto Machín. He worked with numerous orchestras over the next decade, including Cuban trumpeter Vicente Sigler (regarded as probably the first big Latin dance band to perform in NYC during the ’20s), Noble Sissle ‘31-2, and Hi Clark and His Missourians at the Savoy Ballroom. Chick Webb spotted Bauzá while performing with the latter in 1933 and hired him as his lead trumpeter. Webb promoted him to musical director the following year, but he was fired from the band in 1937 or ‘38 after an argument with a club owner or because of the jealously of certain band members (accounts vary). Bauzá helped launch the career of Ella Fitzgerald by introducing her to Webb. In 1936 he married Machito’s sister Estella (c. 1912-1983). During 1937-9 he worked briefly with Don Redman and Fletcher Henderson, then replaced Doc Cheatham in Cab Calloway’s band in 1939. He became a close friend and influence on Dizzy Gillespie (they met in Webb’s band); and gave Gillespie a major break by faking illness to enable Gillespie to substitute for him and be heard by Calloway, who then recruited him. He also introduced Gillespie to Chano Pozo in 1947, helping to change the course of jazz.

In 1941 he joined Machito’s Afro-Cubans, reorganising and expanding the band. While Machito was absent for military service in 1943, Bauzá is said to have been responsible for the genesis of the jazz/Latin fusion called at various times: Afro-Cuban jazz, Cubop and Latin jazz, when he created "Tanga" (meaning: marijuana) during a rehearsal. "Mario, through his efforts married these musics to incorporate as much richness in rhythm and harmony as possible; all bands, therefore that came after the Afro-Cubans were just followers," stated Chico O’Farrill (1921-2001). With Bauzá as musical director, the Machito orchestra redefined the Latin sound as hot Cuban music, as opposed to bands like that of Cugat, which played a polite style for white hotels. Machito described him as the architect of his orchestra. Bauzá may have been the inventor of Afro-Cuban jazz, but many consider that the band’s popular trademark sound was provided by the consistently jazz-inflected arrangements of René Hernández (died: 1977), Machito’s arranger and pianist from 1945 to ‘66. Hernández went on to work with the Tito Rodríguez orchestra and wrote arrangements for many bands on the NYC Latin scene including Eddie Palmieri.

In 1974 Bauzá produced the LP Esa Soy Yo, Yo Soy Asi on Mericana for Graciela (1915-2010; Machito’s sister, and singer with his band from 1943) and co-produced the Grammy-nominated Dizzy Gillespie y Machito: Afro-Cuban Jazz Moods on Pablo in 1975, both with O’Farrill, but split acrimoniously with Machito after 35 years in late ‘75 when he objected to Machito’s son Mario Grillo being moved from bongo to timbales (to replace José Madera): the upshot was that Mario Grillo replaced Bauzá. Bauzá organised a band with Graciela to make the LP La Botanica (Lamp / Coco, 1977), which sold poorly. He continued guesting on LPs such as Rafael Cortijo’s Caballo de Hierro (Coco, 1978) and Típica 73’s Into The ’80’s (Fania, 1981). Bauzá and Graciela’s 1986 project Afro-Cuban Jazz With Graciela, Mario Bauzá And Friends for Caimán Records garnered a Grammy nomination. The twosome also featured on Rica Charanga (Caimán, 1986) by veteran vocalist Rudy Calzado (1929-1911).

Prado gave inconsistent accounts of how and when he started writing mambos. It is posited that he utilised the term already popularised by danzón-mambos performed by the flute and strings orchestra Arcaño y sus Maravillas, to develop a formula for his brass-and-sax jazz-type line-up. He made little impact in Cuba, so went to Mexico in 1948 and worked with various bands before organising another band. He hired the Blanquita de Cuba theatre where he performed the successful show Al Son del Mambo (To The Sound of the Mambo). He teamed-up with the Cuban sonero Benny Moré for highly successful Mexican tours, a notable appearance at Panama carnival and numerous recordings: "El Barbaro Del Ritmo" Mambos by Beny Moré (Tumbao, 1991) collects sides recorded on RCA Victor with Moré between 1948 and ‘50, arguably some of his finest and most authentic recordings.

He made his US debut in May 1951 for a one-nighter with a pick-up band (including Mongo Santamaría) at the Ashland Auditorium for Chicago’s Mexican Youth Center. He took up residence in USA after being kicked out of Mexico in bizarre circumstances. There he recorded for Seeco, United Artists, Epic and other labels, but mostly for RCA. His best-known mambos were numbered: "Mambo No. 5" (1949, said to have been the first major crossover mambo hit), "Mambo No. 8" (both included on Go Go Mambo! ‘92 on Tumbao, a collection of recordings he made in Mexico between 1949 and ‘50 and New York City in ‘51). The mambo gradually became a craze in various Latin countries and a national fad in the USA in 1954. In the USA Latin community the three mambo kings were Machito, Tito Puente and Tito Rodríguez; although the last two also recorded for RCA, Prado became the best known of them all because he had the biggest crossover hit: "Cherry Pink And Apple Blossom White" in 1955. "Cerisier Rose et Pommer Blanc" was published in Paris in 1950, recorded by Prado as "Cerazo Rosa" in 1951 (included in Go Go Mambo!) and again in 1955 for use in the film Underwater with Jane Russell; the arrangement had a slower tempo than Prado’s other mambos and a spectacular trumpet solo by Billy Regis. It was no. 1 in the USA for ten weeks and for two weeks in the UK. He had a second no. 1 in 1958 with "Patricia", a bouncy organ-led jazzish cha cha chá.

In 1930 the group settled for the name La Sonora Matancera (The Matanzas Group). In 1932 José Rosario Chávez "Manteca" replaced Jimagua on timbales. Former Septeto Nacional trumpeter / composer Calixto Leicea replaced the ailing Ismael Goberna (who subsequently died) as first trumpeter in 1935. The same year, Valentín switched to conga and his son Humberto Cané was hired to play tres. Bienvenido Granda (1915-1983) sang lead vocals from the early ’40s to the mid-’50s; he left due to differences with Martínez. Peerless pianist, composer, arranger Lino Frías (died: April 1983) played piano from 1944 to 1976 (he retired due to arthritis), followed by notable New York-based salsa pianist, bandleader, arranger, composer and producer Javier Vázquez (Bubú’s son). Second trumpeter Pedro Knight was added on January 6th, 1944 in order to rival other bands with three trumpets. After marrying Celia Cruz (lead singer ‘50-65) on July 14th, 1962 in Mexico City, he retired from the band on April 30th, 1967. Raimundo Elpidio Vázquez replaced his father Bubú on bass in 1954. Valentín departed in 1947 and Martínez was appointed leader. Manteca left in 1955. Papaíto joined as timbalero in 1960 and trumpeter Alfredo "Chocolate" Armenteros joined Leicea from 1976 to 1980.

Besides Granda and Cruz, over 60 singers performed with La Sonora, including Daniel Santos, Myrta Silva, Miguelito Valdés, Bobby Capó, Vicentico Valdés, Nelson Pinedo, Laíto (Estanislao Sureda), Alberto Beltrán, Carlos Argentino, Leo Marini, Celio González, Willy "El Baby" Rodríguez, Justo Betancourt, Elliot Romero, Yayo el Indio, Roberto Torres, Wuelfo Gutiérrez, Jorge Maldonado, Cali Alemán, Ismael Miranda, Fernando Lavoy and Frankie Vázquez.