Sweeney was born and raised in Shoreham, New York, by an educator father and a municipal government employee mother. He attended Shoreham-Wading River High School and both Tulane and New York University. Before completing his BFA at NYU in 1984, he was cast in the Broadway revival of The Caine Mutiny Court Martial.
He has been married to Ashley Vachon since April 2000 and they have two children.
Sweeney guest starred on television series The Edge of Night and Spenser: For Hire before entering films, where he portrayed a gung-ho Vietnam enlistee in Francis Ford Coppola's Gardens of Stone (1987). Sweeney is perhaps best remembered for his portrayal of Doug Dorsey in The Cutting Edge (1992), Shoeless Joe Jackson in Eight Men Out (1988), or Dish Boggett in Lonesome Dove (1989).
He is currently the voice-over artist for the OWN: Oprah Winfrey Network and his large body of voice-over work includes three seasons of Fox Sports Net's Beyond the Glory and National Geographic Television's Ice Pilots. Past ad campaigns include Bud Light, Lincoln automobiles, Conoco-Phillips, John Deere, Major League Baseball and Coca Cola.