Weston received an Independent Spirit Award nomination for her performance in Dead Man Walking (1995), and also had supporting roles in more than 40 movies, including The Talented Mr. Ripley (1999), In the Bedroom (2001), Hulk (2003), and The Village (2004).
After a recurring role on the ABC daytime soap opera, Ryan's Hope, she joined the cast of the CBS sitcom Alice as Jolene Hunnicutt until the series ended in 1985.
She was nominated for the Independent Spirit Award for Best Supporting Female for her role as Mary Beth Percy in the 1995 crime drama film, Dead Man Walking opposite Susan Sarandon.
In the 2000s she had guest starring roles on Law & Order: Special Victims Unit, Frasier and Desperate Housewives.
From 2010 to 2011 she co-starred alongside Jason Lee and Alfre Woodard in the TNT comedy-drama, Memphis Beat.