Yosvany Terry Cabrera was born in 1972 in Camagüey, Cuba, into a family already steeped in Afro-Cuban music and religion. His father, Eladio "Don Pancho" Terry, was a violinist and chekeré player who led Maravilla de Florida, a well-regarded 1950s charanga ensemble, and the household practiced Afro-Cuban and Haitian vodún traditions that exposed Terry to ceremonial drumming and chant from early childhood. He began on violin at five under his father's instruction and switched to saxophone around age ten or eleven, training further at a boarding arts school in Camagüey before moving to Havana at seventeen to attend the Escuela Nacional de Arte (ENA), graduating in 1992 from a competitive cohort that also included drummer Dafnis Prieto, pianists Osmany Paredes and Roberto Carcassés, and pianist Elio Villafranca.

Out of the conservatory, Terry freelanced around Havana, touring South America from 1992 with the band Diakara behind singer-songwriters Santiago Feliú and Silvio Rodríguez, both figures of Cuba's nueva trova movement rather than jazz musicians. In 1997 he co-founded the group Columna B with Dafnis Prieto, Roberto Carcassés, and Descemer Bueno (later known for Latin pop rather than jazz), touring Spain and playing Havana's jazz festivals. He first traveled to the United States for the Stanford Jazz Workshop in 1995, delayed a year by a visa denial, and there met alto saxophonist Steve Coleman, who became an influential mentor, sharing recordings of Charlie Parker, Sonny Rollins, and John Coltrane and pushing Terry's harmonic vocabulary well past the conservatory curriculum. Terry appears the following year on Coleman's The Sign and the Seal (Transmissions of the Metaphysics of a Culture) (1996), a collaboration between Steve Coleman and the Mystic Rhythm Society and the folkloric group AfroCuba de Matanzas, as well as on trumpeter Jesús Alemañy's founding ¡Cubanismo! album the same year.

Terry relocated permanently to New York in 1999, at twenty-eight, and quickly became a first-call saxophonist and chekeré player across the city's jazz and Latin-jazz circuits, working with Steve Coleman, Dave Douglas, trumpeter Brian Lynch, drummer Jeff "Tain" Watts, pianist Gonzalo Rubalcaba, bassist Rufus Reid, keyboardist Jason Lindner, both New York-based musicians named Avishai Cohen (the bassist and, separately, the trumpeter), and pianist Manuel Valera, alongside touring stints with trumpeter Roy Hargrove's band and pianist Eddie Palmieri's Afro-Caribbean ensembles. He plays alto saxophone on the Brian Lynch/Eddie Palmieri Project's Simpático (2006, ArtistShare), which won the Grammy for Best Latin Jazz Album in 2007, and later contributes saxophone to two of Palmieri's own Ropeadope albums, Full Circle (2018) and Mi Luz Mayor (2018), the latter a tribute to Palmieri's late wife Iraida. He also co-produced and played alto saxophone on Columna B's own Twisted Noon (2001), the band's sole full-length release.

Terry's debut as a bandleader in his own name, Metamorphosis (2005, Ewe Records/Kindred Rhythm), drew on a rotating cast that included his brother Yunior Terry on bass, Pedrito Martinez on percussion, Luis Perdomo on piano, and Mike Moreno on guitar, and established the quintet/septet format he would return to across the decade, including Today's Opinion (2012, Criss Cross Jazz). In December 2006 he received an $85,000 grant from Rockefeller Philanthropy Advisors to research Arará traditions, the Afro-Cuban religious and musical culture rooted in Dahomean (present-day Beninese) practice concentrated in Cuba's Matanzas province; over the following years he studied under the ritual master El Maño (Mario Rodríguez) and, in 2011, was formally initiated into the Sabalú cabildo alongside his brother Yunior. That fieldwork produced New Throned King (2014, 5Passion), a ten-piece ensemble recording built around Arará rhythms and chants set against modern jazz harmony, which earned a 2015 Grammy nomination for Best Latin Jazz Album. The same year, Terry received the Doris Duke Artist Award.

In 2015 Terry joined Harvard University's Department of Music and Office for the Arts as Director of Jazz Ensembles and Visiting Senior Lecturer on Music, a position that made him permanent faculty overseeing the university's two student big bands and its jazz theory and improvisation curriculum; he continues to hold the post, now as Senior Lecturer in Music and Director of Jazz Ensembles. In 2017 he released Ancestral Memories (Okeh Records), a co-led quartet album with French pianist Baptiste Trotignon that set Caribbean, New Orleans, and French-Louisiana material inside a shared jazz language, again featuring Yunior Terry and Jeff "Tain" Watts. He has continued to record and tour into the 2020s, including guest appearances on recent projects such as drummer Raphaël Pannier's 2024 Saint-Louis Jazz Festival (Senegal) live recordings.

Terry's playing joins two vocabularies that most jazz musicians keep separate: the ceremonial Arará and broader Afro-Cuban percussion tradition he absorbed from his father and later researched formally, centered on the chekeré, and the postbop and M-Base-adjacent harmonic language he developed in New York under Steve Coleman's influence. Rather than treating the two as a fusion act, he treats Afro-Cuban ritual rhythm as compositional material in its own right, most explicitly on New Throned King, which sets actual Arará liturgical patterns and chants inside jazz harmony and improvisation rather than using them as decorative color. That scholarly, research-driven approach to Cuban folkloric material, combined with an alto and soprano saxophone sound capable of moving between a dance-band's Latin-jazz drive (heard on his work with Eddie Palmieri) and Coleman-style angular postbop, has made him a distinctive voice as both a leader and one of the more heavily documented Cuban sidemen of his New York generation.