Has two children with his ex-girlfriend, Suzanne Elrod. Son, Adam Cohen, born 1972; and daughter, Lorca Cohen, born 1974.
Cohen had a relationship beginning in the 1970s with the Los Angeles artist Suzanne Elrod, with whom he has two children: a son, Adam, born in 1972, and a daughter, Lorca, born in 1974 and named after poet Federico García Lorca. Adam Cohen began a career as a singer-songwriter in the mid-1990s and fronts a band called Low Millions, while Lorca took the part in her father's tour team during the 2008–2010 world tour as photographer and videographer. She also shot Cohen's video for the song "Because Of" in 2004, while her "Backstage Sketch" was included on Cohen's 2010 DVD Songs from the Road. She has directed and shot video clips for The Webb Sisters and Kamila Thompson. On 2 February 2011, Lorca gave birth to a daughter, Viva Katherine Wainwright Cohen with biological father Rufus Wainwright. Lorca is raising the child.
Cohen has said that "cowardice" and "fear" prevented him from ever actually marrying Elrod. Elrod took the cover photograph on Cohen's Live Songs album and is pictured on the cover of the Death of a Ladies' Man album. She is also the "Dark Lady" of Cohen's 1978 book Death of a Lady's Man. Cohen and Elrod split up in 1979.
"Suzanne", one of his best-known songs, refers to Suzanne Verdal, the former wife of his friend, the Québécois sculptor Armand Vaillancourt, rather than Elrod.