Warchus - whose work also includes the West End show Ghost: The Musical - will replace the Hollywood star as artistic director when he steps down in 2015.
Warchus directed Spacey and Jeff Goldblum in an acclaimed production of Speed-The-Plow at the Old Vic in 2008.
"I couldn't be happier that my friend Matthew Warchus has agreed to take the reins next year," said Spacey.
"Matthew is a thoughtful, intuitive and highly creative director and he has rightly been applauded for his work," Spacey added, praising the "quality and diversity" of his successor's work.
"I know he loves our theatre and I am delighted for our staff, our audiences and for our acting and production communities that he will be its next guardian."
Warchus' range of productions includes an Old Vic revival of Alan Ayckbourn's The Norman Conquests, West End hit Art and a Broadway production of Yasmin Reza's God of Carnage, for which he won a Tony for best direction.
The British director also oversaw the stage adaptation of Lord of the Rings, which ran in London's West End from May 2007 to July 2008 and was reported, at the time, to be the world's most expensive stage production.
In 2010, he directed the acclaimed RSC production of Roald Dahl's Matilda, at Stratford - in which his wife Lauren Ward played the role of Miss Honey. The award-winning musical transferred to the West End in October 2011.
"I am excited and honoured to be following Kevin's galvanising tenure at this wonderful building," said Warchus.
"He has re-established the Old Vic as a globally important theatre and I look forward to continuing to develop it as a hub of invigorating creativity."
Warchus made his first foray in to feature films in 1999, with Simpatico. His film Pride, about a group of gay activists supporting striking miners in 1984, is due to screen at the Cannes Film Festival later this week.
The film stars Bill Nighy, Paddy Considine, Imelda Staunton and Dominic West.
Spacey, who has spent 10 years in London as artistic director of the Old Vic, opens in the one-man show, Clarence Darrow, at the theatre later this week.
Matilda director Matthew Warchus is to take over from Kevin Spacey at London's Old Vic theatre.