In March 1965, when LIFE photographer Bill Eppridge spent time on assignment with Judy Garland's enormously talented daughter, Liza Minnelli was just turning 19 and launching what would prove to be a monumental career of her own. In fact, she was about to debut on Broadway in Flora the Red Menace, in a role that would make her, at the time, the youngest woman in history to win a Tony Award for lead actress. (Also on the show's creative team: composer John Kander and lyricist Fred Ebb, who collaborated for the first time on Flora. They would team up in the future on such Broadway classics as Cabaret — later made into a movie starring Minnelli, of course, in her Oscar-winning role as Sally Bowles — and Chicago.)
Liza allowed Eppridge into her spirited Flora rehearsals, and even invited him to her 19th birthday party in New York City, but in the end LIFE published just one of Eppridge's photos. Here, on her birthday (she was born March 12, 1946, in Hollywood) and in tribute to Minnelli and her enduring career, LIFE.com presents a series of pictures — most of which never ran in LIFE — of a legend in the making.
Liza, of course, was no stranger to show business: her mom was Judy Garland, her dad was Vincente Minnelli, she had performed with her mother at the London Palladium and had done a little work Off-Broadway — but Flora the Red Menace would be her highest profile role to date, the gig that would nudge her out of the shadow of her famous parents and into another spotlight entirely.
Of Minnelli's opening-night performance in Flora, LIFE wrote: "She acted and danced with an awkward, captivating charm, threw out 'What-am-I-doing-here?' looks, sang in a voice that boomed and belted, quivered sweetly, and occasionally got out of control — which only added to her likability. . . . She certainly did look like her mother several rainbows ago. When she sang, there were echoes of Judy, too — the old catches and wavers and throbs that made a song sound as if it were going through a nervous breakdown. But soon Judy's image faded and Liza's came into focus."
In the decades since Eppridge made these intimate shots of the budding star, Liza has performed at the world's most famous concert halls, racked up countless awards for her singing and acting and delighted new generations of fans with scene-stealing appearances in contemporary pop-culture touchstones like Arrested Development. But her early years truly were something special. As Liza herself told LIFE in 1965: "Eighteen is great, but 19 is best of all. That's when you're opening in your first Broadway show."