Law, who presented video artist Elizabeth Price with the £25,000 ($40,260) prize during a ceremony at London's Tate Britain, attacked the coalition's proposed English Baccalaureate (EBacc) for a lack of arts provision.
The actor said that he fears "fewer and fewer schools will provide learning opportunities in the arts, thanks to an act of government cultural vandalism".
"We are blunting our leading edge in the arts," Law added.
Price supported Law's view in her acceptance speech, saying that public funding for the arts had been crucial for her career.
"It's incredibly depressing listening to comments made earlier that a young girl from Luton going to a comprehensive [school] might not be able to imagine being an artist, and might not have the opportunities I've had," she said.
"The idea of young people not making art is a really, incredibly depressing idea... We'll end up with an art that only speaks about a very narrow area of experience."
Price's video installation The Woolworths Choir of 1979 juxtaposes video clips of the Shangri-Las with gothic architecture and news footage of a fatal fire at Woolworths in central Manchester in May 1979.
The Turner Prize is presented annually to a British visual artist under the age of 50, and has become associated with conceptual art across all mediums.
This year's other nominees were performance artist Spartacus Chetwynd, Paul Noble, who specialises in pencil drawings, and filmmaker Luke Fowler.
Each of the runners-up will receive £5,000 ($8,050).