Dunagan began her acting career touring in dinner theater and later performing in regional theaters such as the Asolo Repertory Theatre in Sarasota, Florida, and the Actors Theatre of Louisville before trying her luck in New York City. She made her Broadway debut in the 1979 production of George Bernard Shaw's Man and Superman at Circle in the Square as an understudy for Ann Sachs. Sachs became ill at one point during the show's run and Dunagan filled in for her with great success. This exposure led to her to being signed with International Creative Management.

In 1981, Dunagan performed in the first national tour of Children of a Lesser God. While on this tour, Dunagan visited Chicago for the first time and immediately fell in love with the city. After the end of the tour, Dunagan moved to Chicago and has lived there ever since, performing in more than 30 theaters in the Chicago area. Her work on the Chicago stage has garnered her three Joseph Jefferson Awards and three After Dark Awards.

In 2007-2008, Dunagan returned to Broadway in the Steppenwolf Theatre's transplant production of Tracy Letts' August: Osage County. For her performance, Dunagan won a Tony Award, Theatre World Award a Drama Desk Award and an [(Outer Critics Circle Award]). She went on to portray Violet Weston in productions in London's West End in 2009 and in Sydney, Australia in 2010.

She returned to New York in 2017 to Playwrights Horizons Off-Broadway to perform in the world premiere of "The Treasurer."