The singer-actress collapsed in her Beverly Hills home and was found by one of her 13-year-old sons, spokesman Roger Lane said. The cause of death was not immediately known.
Carter had suffered from diabetes for years, Lane said, and she underwent two brain operations in 1992 to fix aneurysms. She recovered and continued to perform, mostly on stage.
At the time of her death, Carter was in rehearsals at a Long Beach theater for "Raisin," a musical version of "Raisin in the Sun."
Blessed with a big voice and strong stage presence despite her 4-foot-11 height, the heavyset Carter prided herself on her range as a performer, doing musicals and drama as well as comedy.
"She was a pioneer in many ways," said fellow Tony winner Audra McDonald. "She had the ability to be such an incredible comedic musical-theater actress, blow a song all the way to the back of the wall and then come down and be so intimate and beautiful in a ballad."
In addition to her Tony for "Ain't Misbehavin,'" Carter received an Emmy in 1982 for a TV broadcast of the show, which was a revue of Fats Waller songs. Her quietly soulful number "Mean to Me" was a show highlight.
Carter garnered two more Emmy nominations in 1982 and 1983 for "Gimme a Break!," playing a housekeeper to a family headed by a widower who was the town police chief. The show ran from 1981 to 1987. In 1985, an episode was broadcast live — the first for a situation comedy in nearly 30 years.
Carter also played the cruel orphanage operator Miss Hannigan in the 1997 revival of "Annie" and appeared in the movies "The Grass Harp" (1995), "Modern Problems" (1981) and "Hair" (1979).
Carter grew up in Birmingham, Ala., singing in her church choir. She sang on the gospel circuit before moving on to coffeehouses and nightclubs. She longed to sing opera, then aspired to be a belter.
From early in her career until the mid-1980s, Carter struggled with alcohol and drugs, eventually shaking her addictions through a 12-step program.
Carter was married and divorced twice. She is survived by her two sons and a daughter.