Goranson studied acting in Chicago and subsequently landed the role of Becky Conner. It was only the second time she auditioned for a part. She played the character for five seasons, exiting the series in 1992, several episodes into the fifth season, to pursue an education at Vassar College. Her schedule permitted her to continue appearing part-time as Becky. Under her new arrangement, the show's producers and Writers had to take care to write scenes for Becky sparingly. Goranson's exit resulted in Becky's eloping with her boyfriend. Becky would be seen via the phone briefly on some occasions as she now lived out of state. In the next season, although not at the beginning, the producers decided to bring the character back. Goranson was still unavailable, so Sarah Chalke auditioned and won the role.

Since Roseanne, Goranson had several small roles in films and has guest starred in Law & Order: Special Victims Unit, Sex and the City, Damages and Fringe. In her film roles as an adult, she is normally credited as Alicia rather than Lecy. Goranson is currently living in a section of Brooklyn, New York; she played the part of Sandra in The Extra Man which opened in limited release on July 30, 2010.

On April 28, 2017 it was announced that a revival of Roseanne was in the works with most of the original cast and some of the original producers slated to return. In May 2017, it was confirmed that Goranson would be returning in the role of Becky. Chalke will also appear, playing a different role. The rebooted Roseanne series premiered on March 27, 2018 on ABC.