The production will run next summer (dates tbc) helmed by radical Australian director Benedict Andrews, who staged Three Sisters at the venue last year.
The play, which successfully transferred to the big screen in 1951 and made a star of Marlon Brando, centres on fading southern belle Blanche DuBois (Anderson), whose arrival at the home of her sister Stella and her brutish husband Stanley Kowalski upsets their marital dynamic and sets Blanche and Stanley on a violent collision course.
Anderson made her West End debut in 2002 in the world premiere of Michael Weller's two-hander What the Night Is For alongside Roger Allam, for which she won a WhatsOnStage Award. Her subsequent London stage appearances include The Sweetest Swing in Baseball at the Royal Court and A Doll's House at the Donmar Warehouse.
David Lan, the Young Vic's artistic director, told the Daily Mail: "It fell out of the sky. This was Gillian and Benedict working together. They sought each other out, and went through what they'd like to do, and came up with Blanche. With these two people involved there are going to be fireworks on the stage!"
Streetcar is a Young Vic and Joshua Andrews co-production.
Stevenson in Happy Days, Mitchell directs Cherry Orchard
Stevenson's recent stage credits include The Seagull (National Theatre), Duet for One (West End) and The Heretic (Royal Court).
"I am thrilled that the great Juliet Stevenson will make her Young Vic debut as Winnie in Natalie Abrahami's new production of Beckett's masterpiece Happy Days," said Lan. "Winnie is one the greatest and most demanding stage roles ever written."
It's followed by a revival of another modern classic, Arthur Miller's A View From The Bridge, which will run from 4 April to 24 May 2014 directed by Ivo van Hove (Roman Tragedies), making his UK debut.
And maverick director Katie Mitchell will return to the Young Vic to helm a production of Chekhov's The Cherry Orchard, which runs from 10 October to 29 November 2014.
Stephens also adapted the National's multi award-winning production A Curious Incident of the Dog in the Night-time.
Veteran director Peter Brook will return to the Cut with his latest production The Valley of Astonishment, which runs from 20 June to 12 July 2014.
Brook and Marie-Helene Estienne's production "brings together neurological research and Persian verse to transport us from the familiar to the extraordinary on a kaleidoscopic journey into the wonders of the human brain."
And rounding off the season in the main house is the UK premiere of Golem (9 December 2014-17 January 2015), from 1927, who were behind recent hit The Animals and Children took to the Streets.
The only production so far announced for the Maria Studio is Beauty and the Beast (4-21 December), directed by Improbable's Phelim McDermott and starring Mat Fraser and Julie Atlas Muz.