The show, based on Michael Morpurgo's novel, tells the story of Joey, a horse who serves in World War I.
Murpurgo said it was "a show like no other, with puppets at its heart" and that it had "moved millions in London and all over the world".
The last performance at the New London Theatre is scheduled for 12 March.
War Horse, which also won five Tony Awards on Broadway and has been seen in 10 countries, is the most successful play in the National Theatre's history. It was also adapted into an Oscar-nominated film by Steven Spielberg.
When it closes, the London production will have played over 3,000 performances and been seen by over 2.7 million people. A UK tour will begin in autumn 2017.
The National Theatre's artistic director, Rufus Norris, announced the show's closure as he unveiled details of the NT's new season in 2016.
"We could keep it going for longer, we could put a new cast in and drag it out, but our subsidy is to make work here and to reach out to audiences around the country."  Norris told The Guardian.
"We always knew it would come to an end and nobody expected it to last as long as it has."
Directed by Marianne Elliott and Tom Morris, War Horse received its world premiere on 9 October 2007 at the National Theatre, where it played for two seasons before opening at the New London Theatre in March 2009.
Producer Chris Harper said:  'War Horse has wowed audiences around the world, and we are incredibly proud of what the show has achieved over the last eight extraordinary years."
The National Theatre's 2016 productions include:
The West End production of War Horse, a galloping success for the National Theatre, is to close in 2016 after a seven-year run.