He was born in Marietta, Pennsylvania and raised in Roxbury, Massachusetts. Serving in the Army for Korean War duty, Dawson played with the Army Dance Band while stationed at Fort Dix from 1951–1953. During his tenure, Alan explored the post-bop era by performing with pianist Sabby Lewis. After being released from the Army, Alan toured Europe with Lionel Hampton.

Dawson is best remembered as an early teacher of Tony Williams, and Sawyer Brown drummer, Joseph Smyth. Other former students include: Terri Lyne Carrington, Julian Vaughn, Vinnie Colaiuta, Steve Smith, Kenwood Dennard, Gerry Hemingway, Jeff Sipe, Billy Kilson, and many others. He began teaching at Berklee College of Music in 1957. Dawson suffered a ruptured disc in 1975 which led to him halting his touring schedule, to leave Berklee and limit his teaching to his home in Lexington, Massachusetts.

His teaching style emphasized the music as a whole rather than concentrate on percussion alone. He stressed the importance of learning the melody and structure of the tune to better fulfill the role of accompaniment. For this purpose, he had students play over standards while also singing the melody out loud. He constantly strived for balance between musical ideas and strict technique. He was big on rudiments and wrote extensive exercises intended to be practiced with brushes. He believed using brushes with his "Rudimental Ritual" would reduce stick rebound allowing the sense of "picking up" the sticks. While teaching, Alan also maintained a prolific performing and recording career.

Dawson was the house drummer for Lennie's on the Turnpike in Peabody, Massachusetts from 1963 through to 1970. This gig allowed him to perform with a diverse group of jazz artists. Around this time, Dawson was Boston's premier jazz drummer for local acts as well as bigger name touring artists.

Throughout the 1960s Dawson recorded almost exclusively with saxophonist Booker Ervin on Prestige Records. In 1968 Dawson replaced Joe Morello in the Dave Brubeck Quartet and continued until 1972. His performance credits also included stints with Bill Evans, Sonny Rollins, Jaki Byard, Booker Ervin, Sonny Stitt, Dexter Gordon, Lee Konitz, Quincy Jones, Charles Mingus, Tal Farlow and many other top jazz artists.

Dawson's teaching methods have been passed on by many of his former students. Books on his approach have been written by John Ramsay and Osami Mizuno, both former students. Alan Dawson died of leukemia on February 23, 1996.

As leader

• Waltzin' With Flo (1992)

As a sideman

With Dave Brubeck

• Live at the Berlin Philharmonie (Columbia, 1970)
• The Last Set at Newport (Atlantic, 1971)
• We're All Together Again for the First Time (Atlantic, 1973)
• All The Things We Are (Atlantic, 1973–74 )

With Jaki Byard

• Jaki Byard Quartet Live! (Prestige, 1965)
• The Last from Lennie's (Prestige, 1965 )
• Freedom Together! (Prestige, 1966)
• Jaki Byard with Strings! (Prestige, 1968)
• The Jaki Byard Experience (Prestige, 1968)

With Arnett Cobb

• Live at Sandy's! (Muse, 1978)

With Al Cohn

• Play It Now (Xanadu, 1975)

With Sonny Criss

• This is Criss! (Prestige, 1966)
• Portrait of Sonny Criss (Prestige, 1967)
• The Beat Goes On! (Prestige, 1968)

With Booker Ervin

• The Freedom Book (Prestige, 1963)
• The Song Book (Prestige, 1964)
• The Blues Book (Prestige, 1964)
• The Space Book (Prestige, 1964)
• Groovin' High (Prestige, 1963–64)
• The Trance (Prestige, 1965)
• Setting the Pace (Prestige, 1965) – with Dexter Gordon
• Heavy!!! (Prestige, 1966)

With Frank Foster

• Fearless Frank Foster (Prestige, 1965)
• Soul Outing! (Prestige, 1966)

With Terry Gibbs

• Bopstacle Course (Xanadu, 1974)

With Dexter Gordon

• The Panther! (Prestige, 1970)

With Gigi Gryce & Clifford Brown

• Gigi Gryce And His Big Band, Vol. 1 (Blue Note, 1954)

With Lionel Hampton

• Lionel Hampton And His Orchestra Live In Sweden (Century/Stash, 1953)

With Illinois Jacquet

• Go Power! (Cadet, 1966)
• Bottoms Up (Prestige, 1968)

With Hank Jones

• Compassion (Black & Blue, 1978)
• Bluesette (Black & Blue, 1979)

With Quincy Jones

• Jazz Abroad (Emarcy, 1955)

With Eric Kloss

• Grits & Gravy (Prestige, 1966)
• First Class Kloss! (Prestige, 1967)
• Life Force (Prestige, 1967)
• We're Goin' Up (Prestige, 1967)
• In the Land of the Giants (Prestige, 1969)

With Junior Mance

• Harlem Lullaby (Atlantic, 1967)
• I Believe to My Soul (Atlantic, 1968)

With Charles McPherson

• Con Alma! (Prestige, 1965)

With James Moody

• Don't Look Away Now! (Prestige, 1969)

With Houston Person

• Chocomotive (Prestige, 1967)

With Jimmy Raney

• Momentum (MPS, 1975)

With Sonny Rollins

• Live in '65 & '68 (DVD)(NAXOS, 2008)

With Sonny Stitt

• Tune-Up! (Cobblestone, 1972)

With Buddy Tate

• Live at Sandy's (Muse, 1978 [1980])
• Hard Blowin' (Muse, 1978 [1984])

With Phil Woods

• Musique du Bois (Muse, 1974)