Finkel first appeared on the stage at the age of nine, acting for almost thirty-five years in the thriving Yiddish theaters of the Lower East Side as well as performing as a standup comic in the Catskill's so-called Borscht Belt. While never a big star, he worked regularly until the venues began dying out in the early 1960s. He landed a small part in the national touring company of "Fiddler on the Roof" (1965), eventually making his Broadway debut in the role of Tevye the milkman in 1970 in the long-running Broadway company. Between Broadway and the touring company Finkel spent twelve years with the production. Nearly two decades later, Finkel's work in the New York Shakespeare Festival production of a Yiddish classic "Crown Cafe" (1989) earned him an OBIE Award.
Finkel made his film debut in the detective comedy "Off Beat" (1986) and more small roles in films followed. It was his appearance as a shyster lawyer in Sidney Lumet's "Q & A" (1990) that led "Picket Fences" producer/writer David Kelley to cast Finkel in the regular role of public defender Douglas Wambaugh. Though the character was not conceived as Jewish, Finkel brought that quality to Wambaugh, a man who though at times may act a bit deviously, is fundamentally concerned with providing his clients with the best defense possible. For the role Finkel earned a 1994 Emmy Award, announcing at the televised ceremonies that he had waited 51 years for that moment. Following the demise of "Picket Fences", the actor had a regular role on the short-lived revival of "Fantasy Island" (ABC, 1998) and then reteamed with writer-producer David E Kelley to play the spunky history teacher Harvey Lipschultz in "Boston Public" (Fox, 2000- ).