Eli Degibri started on mandolin at seven, through the Jaffa Conservatory of Music's after-school program. Three years later he heard a live jazz concert and switched to saxophone. He picked up back-to-back full scholarships to Berklee College of Music's summer performance program in 1994 and 1995, then a full scholarship to Berklee itself in 1997, at eighteen. After a year there, he was one of six musicians selected nationally for a fellowship at the Thelonious Monk Institute of Jazz, where he studied under Ron Carter, Benny Golson, Jimmy Heath, and Clark Terry, graduating in 1999.

It worked immediately. Straight out of the Institute in 1999, Herbie Hancock brought the twenty-one-year-old into his sextet, and Degibri spent the next two and a half years touring the world playing the Gershwin's World repertoire — a stretch of his career later documented on the 2001 DVD The Jazz Channel Presents Herbie Hancock. That same year, he also turned up on Alon Oleartchik's double live album אולארצ'יק ג'אז (Olarchik Jazz), recorded over three nights, April 13–15, 1999, at Tel Aviv's Camelot Club, sharing the bandstand with pianist Yonathan Avishai and trumpeter Avishai E. Cohen — the trumpeter, not the bassist of the same name who was still years from his own international career — years before either Degibri's or that Cohen's careers took off.

By 2002 he had moved to New York and formed his own quintet with guitarist Kurt Rosenwinkel, pianist Aaron Goldberg, bassist Ben Street, and drummer Jeff Ballard, playing rooms like the Blue Note and the Jazz Standard. That same year he joined drummer Al Foster's band, a chair he held for nearly a decade, through 2011; Foster later singled him out as the only young saxophonist whose playing gave him real joy, not just competence. His debut as a leader, In the Beginning (2003, Fresh Sound New Talent), drew on that same quintet's rhythm section and carried liner notes from Herbie Hancock himself. A duo record with pianist Kevin Hays, One Little Song, followed in 2005, then Emotionally Available (2006, Fresh Sound New Talent) — its nine tracks include a run at Coltrane's "Giant Steps" — and Live at Louis 649 (2008, Anzic Records), a organ trio session with Gary Versace and Obed Calvaire.

2010's Israeli Song (Anzic Records) is the record that put him in a room with Ron Carter on bass, Al Foster on drums, and Brad Mehldau on piano — three of the most recorded names in jazz history, backing an Israeli saxophonist on his own tune. Twelve followed in 2013 on Plus Loin Music, built around his working band with pianist Gadi Lehavi, bassist Barak Mori, drummer Ofri Nehemya, and vocalist/guitarist Shlomo Ydov. That same quartet carried Cliff Hangin' (2015), released on Degibri's own Degibri Records imprint, which earned a five-star review from DownBeat magazine — a score the trade publication, reviewing jazz since 1934, hands out rarely. In 2011, Degibri had also headlined the Red Sea Jazz Festival, Israel's largest annual jazz festival, and that year was named the festival's co-Artistic Director, succeeding bassist Avishai Cohen in the role — a post he held into 2021.

He returned to Hebrew-language material in 2015 with Tea for Two (תה בשניים), a second collaboration with Alon Oleartchik recorded at Pluto Studios, again alongside trumpeter Avishai E. Cohen on one track, this time also featuring singer Achinoam Nini (Noa). In 2018 he released Soul Station, a tribute to saxophonist Hank Mobley built around Mobley's own 1960 Blue Note album of the same name, with a new quartet of pianist Tom Oren, bassist Tamir Shmerling, and drummer Eviatar Slivnik. That same year, he became head of Jazz Studies at the Jerusalem Academy of Music and Dance, meaning the players training there now are learning from him directly.

His most recent album, Henri and Rachel (2020, Degibri Records), is named for his parents. His father Henri died in the fall of 2020; his mother Rachel lives with Parkinson's disease and dementia. Degibri has described the years of piano lessons and counterpoint study behind the record as having opened up how he hears and writes music. Along the way he has also played the UN General Assembly for UNESCO's first International Jazz Day in 2012 and performed at the White House for President Obama in 2016, and has picked up Israel's Prime Minister's Award for Jazz Composition and the Landau Award for Jazz Performance.

Eight albums as a leader since 2003, a decade-long apprenticeship split between Herbie Hancock's sextet and Al Foster's band, and, since 2018, a faculty post training the next generation of Israeli jazz musicians — Degibri's career runs from sideman-to-legends to institution-builder without ever really pausing as a working saxophonist.

Degibri has named his three primary tenor saxophone influences directly: Stan Getz, John Coltrane, and Sonny Rollins, describing them as musicians who "accompany" him, present in his playing the way his parents are present in his life. Hank Mobley sits alongside them as a specific, career-defining influence — Mobley's 1960 album Soul Station was enough of a touchstone that Degibri built an entire 2018 record around it. Reviewers describe him as an exceptionally melodic improviser with a big, bold tenor tone (Bill Milkowski, JazzTimes), and on the Soul Station tribute itself, critics noted his tone runs slighter and more pointed than Mobley's round, warm original, with a more forceful, insistent attack across the record.

That combination — a hard-bop and post-bop vocabulary built on Getz, Coltrane, Rollins, and Mobley, delivered with a compositional streak that has drawn praise from Herbie Hancock himself — sits at the center of his own catalog, whether he's fronting a working quartet of young Israeli musicians or sitting in with Ron Carter and Brad Mehldau. Four years of classical piano and counterpoint study, referenced around Henri and Rachel, also feeds his writing directly.