Braxton Cook was born in Boston on March 27, 1991, and grew up mostly in Silver Spring, Maryland, a suburb of Washington, D.C. His father steered him toward the saxophone at the age of five, and the records played at home shaped his ear. At Springbrook High School he studied with Paul Carr, sat in at local gigs, and in 2009 was one of 30 students chosen nationally for the Grammy jazz ensemble. He went to Georgetown University for two years as an English major with a concentration in African American Studies, was named a YoungArts finalist in 2010, and decided during that time to make music his career.

In 2011 he transferred to the Juilliard School on a full scholarship and studied jazz saxophone with Ron Blake and Steve Wilson. He was also a semi-finalist in the 2013 Thelonious Monk International Jazz Saxophone Competition. In his first year at Juilliard he attended a concert by Donald Harrison and met Harrison's nephew, the trumpeter Christian Scott aTunde Adjuah, now known as Chief Xian aTunde Adjuah. Shortly afterward Cook became a long-term member of Scott's touring band. Alto saxophone is his chair on Scott's Stretch Music (2015), Diaspora (2017) and The Emancipation Procrastination (2017); the last was nominated for Best Contemporary Instrumental Album at the 61st Grammy Awards.

His own recording career began in 2014 with the self-released EP Sketch. Braxton Cook Meets Butcher Brown followed on August 14, 2015, a collaboration with the band Butcher Brown that drew notice from critics and from DJs in Britain, Gilles Peterson among them. He signed to the independent label Fresh Selects in 2016, and Somewhere in Between (2017) was the first record to feature his singing as well as his saxophone, blending jazz with the production habits of R&B, hip-hop and dance music. The Fader called him a "jazz prodigy" in its coverage of the album. No Doubt (2018) reached the top of the jazz albums streaming charts and passed a million streams in its first month, which brought festival bookings at Tourcoing Jazz, So What's Next and Ronnie Scott's in London, then the North Sea Jazz Festival and the Montreux Jazz Festival in 2019.

Alongside his own band he became a busy sideman. He has toured with trumpeter Marquis Hill, Solange and Rihanna, has appeared with the Christian McBride Big Band, and has recorded with Mac Miller, Lion Babe, Middle Clash and Lauren Desberg. In 2018 he appeared in a Vox video on John Coltrane's "Giant Steps" titled "The most feared song in jazz, explained," which won an Outstanding New Approaches: Arts, Lifestyle and Culture award at the 40th News and Documentary Emmy Awards; Cook received an Emmy certificate for his part in it. In 2021 he was featured on "We Will Rise," a Blue Note release by the duo Blue Lab Beats.

Fire Sign, a quintet record mixing contemporary jazz, electronica and polished soul, came out on March 27, 2020, his 29th birthday. In May of that year, with no band available, he played an NPR Tiny Desk home concert from his house in New Jersey, supporting his vocals and saxophone with loops of guitar, and including his "Hymn (for Trayvon Martin)." He signed with Nettwerk in 2022. That September he released the Black Mona Lisa EP, with Marquis Hill on the single "The Same," and guested on the Stones Throw single "U Already Know" by Kiefer, while Nettwerk reissued Fire Sign on LP and CD. His former Juilliard roommate, the producer Jahaan Sweet, used a vocal sample of Cook's in making "Lavender Haze," the opening track of Taylor Swift's Midnights (2022), and Cook is credited as an additional producer on it. The two had earlier worked together on "Tryna Be" for Giveon's Give or Take (2022).

Who Are You When No One Is Watching? (Nettwerk, February 24, 2023) is an hour-long set that includes Masego on "90s" and Orlando Watson on "The Answer," and opens with "MB (For Ma'Khia Bryant)," a tribute to the teenager killed by police in Columbus, Ohio, in 2021, whose melody he improvised over loops from the English drummer Moses Boyd. On February 3, 2025 he returned to NPR to lead his own band at the Tiny Desk for the first time, after earlier visits as a sideman with Scott in 2015, Tom Misch in 2018, Phony Ppl in 2019 and Amaarae in 2023. His most recent album, Not Everyone Can Go, followed on Nettwerk on August 29, 2025.

Cook is a jazz-trained alto saxophonist who also sings, writes, loops and produces, and a large part of his catalog is closer to R&B and soul than to straight-ahead jazz. AllMusic describes his work as an integration of contemporary and post-bop jazz with soul, funk, rap and gospel, and files it under Contemporary Jazz, Electric Jazz and Alternative R&B. On his own records he often sings lead, writes songs in the R&B tradition and works with hip-hop and R&B producers, so the saxophone solos arrive inside songs built like soul records. The Grammy Recording Academy's site, reviewing a living-room performance of his tune "Indie," described his saxophone tone as light and conversational, with the phrasing of an improvising jazz player.

The jazz lineage is explicit. His long apprenticeship in Scott's band places him in the modern post-bop and fusion wing of New Orleans jazz, AllMusic lists Scott, Chief Xian aTunde Adjuah, Robert Glasper, Terrace Martin, Gary Bartz and Kenny Garrett among related artists, and the Coltrane video helped carry his name beyond jazz circles. NPR notes that his composition "Zodiac" was inspired by the pianist Mary Lou Williams, and the track "Black Mona Lisa" takes its cue from a Nat King Cole song. His songs take on family, faith and current events. He plays a 1923 Conn New Wonder alto saxophone. Writers at outlets including The Fader, The New Yorker and JazzTimes have called him a prodigy, and Ebony named him one of the top five jazz artists to watch in 2018. The Subgenres above are the closest canonical jazz labels; the R&B and soul side of his work has no equivalent in that list.