Eugene Joseph Wright was born in Chicago on May 29, 1923, the son of Mayme (Brisco) Wright, who played piano, and Ezra Wright. He studied cornet in high school and, in his early twenties, led a 16-piece band called the Dukes of Swing; Discogs dates the group to 1943 through 1946 and lists two singles for The Aristocrat of Records, "Pork'n Beans" and "Dawn Mist" (1948), and a 1949 coupling with the vocal group the Dozier Boys. He taught himself the double bass, took private lessons from Paul Gregory and others in his early thirties, and made Walter Page, the bassist of Count Basie's original rhythm section, his idol.

His earliest documented sideman work comes from 1947: a Mercury sextet date led by tenor saxophonist Gene Ammons with the pianist Junior Mance, and a Savoy session for baritone saxophonist Leo Parker in Detroit on October 4, 1947. In the late 1940s he also played with the Count Basie Orchestra, and Discogs credits him on a 1997 release of the band's September 1948 Royal Roost broadcasts, which also feature Clark Terry, Dickie Wells, Wardell Gray, Dinah Washington and Jimmy Rushing. Discogs dates his association with Ammons from 1946 to 1951. He played on saxophonist Sonny Stitt's Prestige quartet sides from 1950 to 1953, with Mance and drummer Art Blakey, and on an Arnett Cobb orchestra session recorded in New York on November 20, 1951. From 1952 to 1955 he was in Buddy DeFranco's group, touring Europe with the clarinetist and playing on his 1954 Norgran sessions with the pianist Sonny Clark and drummer Bobby White. In 1955 he joined the Red Norvo trio and toured Australia with it. On the West Coast he worked in vibraphonist Cal Tjader's groups, playing on part of Tjader Plays Tjazz (sessions of June 6, 1955) and on Jazz at the Blackhawk, taped at San Francisco's Blackhawk on January 20, 1957, with Vince Guaraldi on piano and Al Torre on drums. He was also featured in a film short with the bandleader Charlie Barnet.

When bassist Norman Bates left Dave Brubeck's quartet in 1958, drummer Joe Morello suggested Wright, who auditioned at Brubeck's home in Oakland, California, playing "Brother, Can You Spare a Dime?" with the pianist and winning the job. His first test came on February 5, 1958, at East Carolina College in Greenville, North Carolina, where the dean of student affairs objected to a Black musician onstage; Brubeck refused to play without him, and the school agreed to let the quartet perform only on the condition that Wright stay in the background. Brubeck defeated the condition by telling Wright his microphone was broken and that he would have to take his solo at the announcement microphone at the front of the stage. Wright went on the quartet's 1958 State Department tour of Poland, Turkey, Iran, Iraq, India, Afghanistan, Pakistan and Ceylon, and became its permanent bassist in 1959. Brubeck turned down a South African tour in October 1958 because the government would not admit Wright, and in March 1959 he cancelled a University of Georgia concert rather than replace him. In January 1960 he cancelled a Southern college tour of 25 dates after 22 of the schools refused to accept an integrated band. In 1964 the quartet played to an integrated audience at the University of Alabama's Foster Auditorium in Tuscaloosa despite Ku Klux Klan picketing and threats.

The music he made in those years is the center of his legacy. With Brubeck, alto saxophonist Paul Desmond and Morello he recorded Time Out on June 25, July 1 and August 18, 1959, released that December; it was the first jazz album to sell a million copies, and "Take Five" reached number 25 on the Billboard pop chart. About 30 albums with the quartet followed, among them Southern Scene (1960), where he is featured on "Happy Times," Time Further Out (1961), At Carnegie Hall (1963), Time Changes (1964), and Jazz Impressions of Japan (1964). In September 1961 he recorded Dave and Iola Brubeck's jazz musical The Real Ambassadors with Louis Armstrong, Carmen McRae and Lambert, Hendricks & Ross, which was staged at the 1962 Monterey Jazz Festival. Away from the quartet he made his own LP, The Wright Groove, issued by Philips in 1962 or 1963 and recorded, according to Discogs release notes, in Wellington, New Zealand, with baritone saxophonist Laurie Lewis, pianist Lew Campbell and drummer Don Branch. He also played bass on several of Desmond's RCA Victor quartet albums with guitarist Jim Hall, among them Take Ten (1963) and Bossa Antigua (1965). Brubeck disbanded the group at the end of 1967, and its last record, The Last Time We Saw Paris, was taped at the Salle Pleyel in Paris on November 13, 1967.

In 1969 and 1970 Wright led his own ensemble on a tour of historically Black colleges, and from 1971 to 1974 he was bassist in Monty Alexander's trio, recording Here Comes the Sun (June 1971), We've Only Just Begun (December 1971) and Perception! (1973) with him. In the 1970s he also did studio work for film and television. He reunited with the Brubeck quartet on several occasions, including the 25th anniversary reunion album recorded in March 1976 and released on Horizon in 1977 and, according to the Brubeck estate, a 1988 joint state dinner in Moscow organized by Reagan and Gorbachev. He later headed the jazz department at the University of Cincinnati, led the advisory board of the jazz division of the International Society of Bassists, and mentored younger bassists in Los Angeles; he played private parties until 2016 and gave lessons until about 2018. Known across the jazz world as "The Senator," Wright passed away at an assisted living facility in the Valley Glen neighborhood of Los Angeles on December 30, 2020, at the age of 97, the last surviving member of the classic Brubeck quartet.

Wright played with a firm, swinging time that critics traced to Walter Page and to the Kansas City tradition, an apparent mismatch for Brubeck's cool, meter-bending music that worked because he never wavered. Brubeck wrote that Wright's steady lines grounded the group and freed him, Desmond and Morello to try other tempos and counter-rhythms. The biographer Philip Clark described a light-fingered bassist who could swing hard and had a soft, spongy sound that gave Time Out and intricate pieces such as "Three to Get Ready" something of a chamber-music quality, and Chris Brubeck called him an egoless player who nevertheless soloed well. His name is also tied to a principle: the quartet's refusal to perform without him at segregated venues, which Brubeck cast as part of his fight for equality and which Wright, a quiet public figure, left largely to the bandleader to speak for. His long list of employers, from Count Basie Orchestra and Gene Ammons to Cal Tjader, Vince Guaraldi and Monty Alexander, shows a bassist at home in swing, bebop, Latin jazz and the cool West Coast style.