Roberto Cossa was born in Buenos Aires, Argentina, and raised in the quiet residential borough of Villa del Parque. He first performed in theatre at the age of 17 and, in 1957, he and friends founded the San Isidro Independent Theatre. An admirer of Fidel Castro, he worked secretly as a local correspondent for Cuba's state-owned press agency, Prensa Latina, between 1960 and 1970. Cossa produced his first play, Nuestro fín de semana ("Our Weekend") in 1964. The Neo-realist work earned him numerous Argentine drama prizes and secured his reputation in the field. Contributing to the cultural sections of mainstream Argentine newsdailies such as Clarín, La Opinión and La Nación between 1971 and 1976, Cossa avoided direct political references in his work. One exception to this was his 1970 play El avión negro ("The Black Plane"), a commentary on exiled populist leader Juan Perón's 1964 attempt to return to Argentina.

Following a period of a certain creative dearth, Cossa premiered La nona ("Grandma") in 1977. His most successful play, La nona represented a turn towards the grotesque in which the protagonist, a hundred-year-old Italian Argentine grandmother, burdens her working-class family with her senile dementia and ravenous appetite. La nona, still performed in Buenos Aires and elsewhere, remains among the most recognizable plays in Argentine theatre and was adapted into a film version in 1979.

The climate of repression that prevailed in Argentina during its last dictatorship eased somewhat in 1980 as General Jorge Videla prepared to transfer power to General Roberto Viola, an advocate for increased, if limited, artistic freedom. Playwright Osvaldo Dragún seized the opportunity to organize an Teatro Abierto ("Open Theatre") movement, calling on Cossa and fellow playwrights Luis Brandoni, Jorge Rivera López and Pepe Soriano, as well as receiving support from prominent intellectuals such as Nobel laureate Adolfo Pérez Esquivel and writer Ernesto Sábato. Refurbishing a former Buenos Aires sparkplug factory into the Picadero Theatre, they premiered their first festival on July 28, 1981, featuring Cossa's Gris de ausencia ("Pale of Absence") among the evening's repertoire. During an August 6 performance, however, three fire bombs were set off in the theatre, forcing the Open Theatre to relocate (The Picadero was reopened in 2001).

Cossa's successful adaptation of La nona into film encouraged him to provide screenplays for his play El arreglo ("The Deal") and No habrá más penas ni olvido ("Dirty Little War") in 1983. Both became well-known Argentine films memorable for their bold indictments of the problems of corruption and revenge in Argentine society. His theatre works became more prolific following democratic elections in 1983 and included Ya nadie recuerda a Frederic Chopin ("No One Remembers F.C."), a study in frustrating exile, De pies y manos ("On Hands and Feet"), a realist look into the effect of the dictatorship on one family, and Los compadritos ("The Poseurs"), a controversial review of the events surrounding the 1939 sinking of the Graf Spee.

Cossa, in 1987, premiered what would become his most successful work since La nona, Yepeto ("Gepetto"). The character study sets an aging drama professor in a love triangle with two students, one a young lady leading a charmed life with whom he becomes infatuated and the other an impetuous young man whom the teacher feels he must rescue from himself, despite his jealousy for the unabashed youth. The tension made Yepeto a hit in Argentine theatres and, starring La nona great Ulises Dumont, it ran for about 5000 performances in the 1990s before its 1999 film release.

Conferred the National Theatre Prize Argentina and the Public and Critics Prize of Spain, Cossa was appointed President of the General Society of Argentine Authors in 2007.