The "Sound of Music" star, who was appearing in the musical "My Fair Lady" at the time, found her fellow Brit sobbing in a hotel elevator during her run in the "New Faces Of '56 revue" at the Ethel Barrymore Theatre and offered to help.
"I wasn't really paid very much," Smith told Gyles Brandreth during a recent Fane online event. "In fact, I think it was $16 a week... And I had been booked into a hotel which of course I could not pay for. But I was rescued by Julie Andrews."
"I had been put in this hotel and Julie was there with (set and costume designer) Tony Walton. I was going up in the elevator and weeping, which is what I did all the time because I was in such distress about the whole thing. They saw that I was weeping and in a terrible state and they asked me to have tea with them, which I did."
"Then Julie's agent rescued me. I didn't realise that I would have to pay (labour union) Equity and my agent and my London agent."
Despite winning two Oscars and five BAFTA awards, Maggie Smith sometimes felt inferior to her fellow Dames including Judi Dench.
"Of course, you learn all the time. I learnt so much from Judi, from all these people that I work with. They all seemed to be so at ease with it," she said. "I always felt… And I know what they call it, Imposter (syndrome). I felt that. I still feel it. I look back and I can't believe that a lot of it has happened."