Tickets for the extension go on sale next Wednesday (16 July) at 10am.
The venue has also announced it will run a "day seat lottery" for every performance. Names will be taken at the box office in person at 5pm with winners announced at 5.30pm for each evening's performance (1pm and 1.30pm for matinee performances).
All tickets in the lottery will cost £20 with £10 concession for under 26s. Two tickets can be won per person in the Streetcar lottery.
Directed by Benedict Andrews (Three Sisters), the production opens on 28 July 2014 (previews from 23 July).
Tennessee Williams' 1947 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.
The cast at the Young Vic also includes: Ben Foster, Vanessa Kirby, Corey Johnson, Clare Burt, Lachele Carl, Branwell Donaghey, Otto Farrant, Nicholas Gecks, Troy Glasgow, Stephanie Jacob and Claire Prempeh.