Dame Angela, 89, won best supporting actress for the role of eccentric medium Madame Arcati in Blithe Spirit.
The Kinks-inspired musical Sunny Afternoon and drama A View From the Bridge both won multiple awards.
Hosted by Lenny Henry, the Oliviers ceremony took place at London's Royal Opera House on Sunday night.
Accepting her statuette, Dame Angela said: "All these years of waiting.  I am so infinitely grateful to have this baby in my hands."
The Murder, She Wrote star, who was born in east London in 1925, recalled how she started her stage career in "a lovely play" with Dame Peggy Ashcroft, the title of which she couldn't recall.
"I can't remember a lot of things these days - except I can remember my lines," she joked.
She added: "Here I am creeping up to 90 and feeling like a million dollars because I'm in London in this magnificent hall with all you - my roots, where I began."
The actress's star turn in Blithe Spirit took place at the Gielgud Theatre where her mother, actress Moyna Macgill, made her debut on the same stage in 1918 - when the theatre was known as The Globe.
The best actor prize went to Mark Strong, for his brooding performance in Arthur Miller's A View from the Bridge, which enjoyed a sell-out run at Wyndham's Theatre and might soon transfer to Broadway.
The role of tragic hero Eddie Carbone had lured Strong back to the stage after 12 years in film.
"What's been amazing about doing this play is the young people who that came to see it who all wanted to talk about what they were seeing," he said.
"Thousands of years have gone by and we still have this thing called live theatre - and the reason is that we need to be able to compare ourselves to what we see up there and judge ourselves as human beings."
The play, originally staged at London's Young Vic, also won best revival and best director award for Ivo Van Hove.
Penelope Wilton won best actress for Taken at Midnight in which she played the mother of a young German lawyer imprisoned by the Nazis.
Accepting her award, she said it was a story "that needed to be told again".
"It has a resonance today even though it was about a man who died in 1937 - it's about the importance of democracy and freedom of speech."
Joe Penhall's play Sunny Afternoon, which transferred to the West End from London's Hampstead Theatre, was crowned best new musical.
Stars John Dagleish and George Maguire - who play The Kinks brothers Ray and Dave Davies - were awarded best actor and best supporting actor in the musical category.
Ray Davies received a special outstanding achievement award for the Sunny Afternoon score.
Picking up his award, Davies described The Kinks as "four of the unlikeliest pop stars you've ever seen".
He said: "When you write songs you write about people.  Without people we have no plays we have no films. People are the source of my material.
"The next time you are sitting in a park somewhere and you see someone like me looking at you, don't phone the police. I'm just writing about you."
Beautiful - The Carole King Musical took two awards for its stars Katie Brayben and Lorna Want.
Modern history tale King Charles III won British playwright Mike Bartlett one of his two awards, for best new play.
He also won the award for outstanding achievement in an affiliate theatre for Bull at The Maria at Young Vic.
Meanwhile Nathaniel Parker's portrayal of Tudor King Henry VIII in Wolf Hall and Bring Up the Bodies won him best supporting actor, with the play also honoured for its costume design.
Wicked won the This Morning Audience Award, the only prize of the night voted for by the public, beating Billy Elliot, Matilda and Jersey Boys.
Despite leading the nominations with nine, Memphis the Musical won just two awards, for theatre choreographer and sound design.
Two special awards were also handed out, to groundbreaking dancer Sylvie Guillem and Kevin Spacey.
Dame Judi Dench presented Spacey with his award, given in recognition of his tenure as artistic director at The Old Vic theatre.
"`I love that theatre more than I can begin to express to you," he said, before he closed the ceremony with a performance of the Simon and Garfunkel song Bridge Over Troubled Water, which he sang with singer and Memphis star Beverley Knight.
The full list of 2015 Olivier Award winners:
Best actor
Mark Strong for A View from the Bridge at Young Vic and Wyndham's Theatre
Best actress
Penelope Wilton for Taken At Midnight at Theatre Royal Haymarket
Best revival
A View from the Bridge at Young Vic and Wyndham's Theatre
Best director
Ivo Van Hove for A View from the Bridge at the Young Vic and Wyndham's Theatre
Best new play
King Charles III at Almeida Theatre and Wyndham's Theatre
Best actor in a supporting role
Nathaniel Parker for Wolf Hall and Bring Up the Bodies at Aldwych Theatre
Best actress in a supporting role
Angela Lansbury for Blithe Spirit at Gielgud Theatre
Best actor in a musical
John Dagleish for Sunny Afternoon at Hampstead Theatre and Harold Pinter Theatre
Best actress in a musical
Katie Brayben for Beautiful - The Carole King Musical at Aldwych Theatre
Best new musical
Sunny Afternoon at Hampstead Theatre and Harold Pinter Theatre
Outstanding achievement in music
Ray Davies for Sunny Afternoon at Hampstead Theatre and Harold Pinter Theatre
Best actor in a supporting role in a musical
George Maguire for Sunny Afternoon at Hampstead Theatre and Harold Pinter Theatre
Best actress in a supporting role in a musical
Lorna Want for Beautiful - The Carole King Musical at Aldwych Theatre
This Morning Audience Award
Wicked at Apollo Victoria Theatre
Best new comedy
The Play That Goes Wrong at Duchess Theatre
Best musical revival
City of Angels at Donmar Warehouse
Best lighting design
Howard Harrison for City of Angels at Donmar Warehouse
Best sound design
Gareth Owen for Memphis The Musical at Shaftesbury Theatre
Best entertainment and family
La SoirÃ©e at La SoirÃ©e Spiegeltent
Best costume design
Christopher Oram for Wolf Hall and Bring Up The Bodies at Aldwych Theatre
Best set design
Es Devlin for The Nether at Duke of York's Theatre
Best new dance production
32 Rue Vandenbranden by Peeping Tom at Barbican
Mats Ek's Juliet And Romeo by Royal Swedish Ballet at Sadler's Wells
Outstanding achievement in dance
Crystal Pite for her choreography in the productions of The Associates - A Picture of You Falling, The Tempest Replica and Polaris at Sadler's Wells
Best new opera production
The Mastersingers of Nuremberg at London Coliseum
Outstanding achievement in opera
Richard Jones for his direction of The Girl of the Golden West, The Mastersingers of Nuremberg and Rodelinda at London Coliseum
Outstanding achievement in an affiliate theatre
Bull at The Maria at Young Vic
Best theatre choreographer
Sergio Trujillo for Memphis The Musical at Shaftesbury Theatre
Special awards
Kevin Spacey and Sylvie Guillem
Dame Angela Lansbury said she felt "like a million dollars" after winning her first Olivier Award for her first West End role in nearly 40 years.