Elchin Shirinov was born in Baku in 1982, into a family already full of music but not of pianos. His two brothers studied traditional Azerbaijani instruments at the Baku State Conservatoire, and it was only after they began taking piano lessons at the school that their father bought an instrument for the house. Elchin, the youngest, started playing it as well. The turn toward jazz came from a cassette one of his brothers handed him, of Oscar Peterson and Bill Evans, which he wore out by repetition. He is a rarity among internationally working jazz pianists in never having attended a music school at all: he taught himself first, took private instruction later, and built his vocabulary by transcription and listening rather than through a conservatory programme. The teachers he names are the Baku jazz pianist Vagif Sadikhov and, subsequently, four working pianists he studied with directly: Kevin Hays, Aaron Goldberg and Jean-Michel Pilc in New York, and Yakov Okun in Moscow.

His working life began in 2008, when he started playing at the Jazz Center in Baku with a trio of local musicians and began appearing on the Azerbaijani festival circuit. Azerbaijan has a jazz culture older and deeper than its size suggests, and Shirinov came up inside it: he toured with the singer Sevda Alekperzade and joined the quartet of the saxophonist Rain Sultanov, playing festivals in Georgia, Azerbaijan and Malaysia, and led his own trio at the Baku Jazz Festival across several editions between 2010 and 2016. Poland became a second base early. He spent years at the Voicingers festival, where his job was accompanying singers, and in 2013 he toured the country with the vocalist Grzegorz Karnas alongside bassist Michał Jaros and drummer Sebastian Frankiewicz. The same year took him to the Montreux Jazz Festival with Alekperzade and to the Jazzy Colors festival in Paris with his own trio.

2014 is the year his career stopped being a regional one. He began working with players from the London scene, first in Cannes with drummer Jon Scott and the Italian bassist Andrea Di Biase, then in Paris and Athens with Sam Lasserson. He appeared at the Baku Jazz Festival that year with the American rhythm section of Ben Street and Jeff Ballard, and at the Vomex festival in Santiago de Compostela with Street and Adam Cruz. On 26 November he was at Porgy & Bess in Vienna in Rain Sultanov's band with bassist Yasuhito Mori and drummer Peter Nilsson, a concert that was recorded and released. By 2015 the Di Biase and Dave Hamblett trio was his main vehicle, playing the Vigadó in Budapest, the Mugham Festival in Baku, the Sunside in Paris, and venues in Brussels and Moscow. Its German debut was at the Düsseldorfer Schauspielhaus, followed on 19 May 2015 by the Gesellschaftshaus in Frankfurt's Palmengarten, as part of a nationwide Azerbaijani cultural series; the reviewer Stefan Pieper, writing in JazzZeitung, described the Düsseldorf concert as evidence that Shirinov belonged among the best jazz pianists then working. In April 2016 the same trio played the Jazzahead trade fair in Bremen. He has also worked with the Mauritian bassist Linley Marthe and the drummer Eric Harland.

His own recordings began in this period and stayed deliberately close to Azerbaijani material. Maiden Tower (2017), named for the medieval landmark in Baku's old city, was cut in Hungary with a co-billed group of Hungarian players: Márton Fenyvesi on guitar, Mátyás Szandai on double bass and András Dés on percussion. Waiting, first issued digitally in December 2018 and later pressed as a Japanese CD on Somethin' Cool, is the album he counts as his leader debut, made with Di Biase and Hamblett. Its programme sets his own pieces beside a traditional Azerbaijani song, "Sari Gelin", and two adaptations from the Azerbaijani concert repertoire: the Waltz from Gara Garayev's ballet Seven Beauties and a theme from Uzeyir Hajibeyov's operetta O Olmasin, Bu Olsun. On 27 May 2018, by decree of the President of the Republic, he was named an Honoured Artist of Azerbaijan.

The Avishai Cohen years, which made him internationally visible, ran to October 2022; sources differ on whether they began in 2018 or 2019, though the first recording is from the spring of 2019. Shirinov held the piano chair in the bassist's working trio across three albums and a great deal of touring, taking the group to Japan, Hungary, the United Kingdom, France, Germany, Portugal and home to the Heydar Aliyev Center in Baku. Arvoles (2019) put him with drummer Noam David in a group extended by flute and trombone on half its tracks, and includes a Cohen composition titled "Elchinov". Two Roses (2021) set the trio, with Mark Guiliana on drums, against the Gothenburg Symphony Orchestra. Shifting Sands (2022), with the young Israeli drummer Roni Kaspi, was John Fordham's jazz album of the month in the Guardian, at four stars, in a review that called Shirinov "the thoughtful but forceful Azerbaijani musician who played on both Two Roses and Arvoles". In November of that year All About Jazz's readers placed him 180th in a poll of the two hundred favourite living jazz pianists. Guy Moskovich took over the piano chair in Cohen's trio at the end of 2022.

Since leaving the trio he has worked mainly from Britain. A duo with the Scottish singer Louise Dodds, begun in late 2022, produced Two Hours After Midnight (Loch Tay Records, 2023), a programme built partly on the poetry of Robert Burns, and the pair played the Edinburgh Festival Fringe that August. He has continued to appear as a leader and a guest across Europe, on Rain Sultanov's Mr. Joe (Live) (2025) alongside Linley Marthe and Paco Sery, and on a run of his own singles from 2024 onward, several of which return to Azerbaijani subjects: "Khojaly", "Arazbari" and "My Dear Baku", the last with Panagiotis Andreou and Henry Cole. He is currently raising founding support for a jazz club of his own in the Covent Garden and Soho district of London.

Shirinov's distinguishing material is mugham, the Azerbaijani modal and improvisatory tradition, which he treats as a working harmonic and melodic resource rather than as decoration; AllMusic describes the result as the folk and classical musics of his country crossed with a complex but smooth strain of post-bop. What that sounds like in practice is a player who will take an Azerbaijani folk theme such as "Durna" at furious speed, full of eastern intervals, microtonal inflection and insistently repeated figures, and then resolve it into something unmistakably post-bop; a London reviewer describing exactly that performance noted the staccato answers and the escalating tempo it drew out of him. The lineage he claims is orthodox American, and studied by ear rather than in a classroom: Art Tatum, Charlie Parker, Bud Powell, Thelonious Monk, John Coltrane, Bill Evans, Miles Davis, Wayne Shorter, Herbie Hancock and Brad Mehldau. In an ensemble he is valued for restraint. Michael Jackson, reviewing the Avishai Cohen Trio at Ronnie Scott's for DownBeat in 2022, credited him with a command of Middle Eastern and Sephardic ornament and with succinct playing and uncluttered time; the Guardian's four-star notice for Shifting Sands found his "upturning elisions on resolving figures" closer to the phrasing of a singer than of a pianist. It is a style built for accompaniment as much as for solo display, which is what three years in one of the most exposed piano chairs in European jazz asked of him.