The Hamlet star fell ill with appendicitis during the run at London's National Theatre and Hopkins was asked to step in to take over from Olivier for four shows.
The Silence of the Lambs actor admits he was wracked with nerves throughout the performance, but he managed to win over the audience and earn their applause.
He says, "I was 28 and I was in a play called The Dance of Death and he (Olivier) was taken ill and I thought they (producers) would cancel the show and they said, 'No, you're going on', and his wardrobe fitted me with same ties and it was a terrifying experience.
"But, the audience showed up and I remember halfway through the show I was looking out at the audience thinking, 'I better go home, go home, go home, and then another part of my brain kept looking and I did the performance and they gave me a standing ovation."
Olivier was released from hospital in time to attend Hopkins' last show and he made sure to reach out to his understudy to praise his efforts.
Hopkins adds: "Olivier came out of the hospital in his bathrobe just to watch and he phoned me the next morning and said, 'Well done'. It was great, it was extraordinary, I couldn't believe I was doing it."