Price, who grew up in Luton, had originally forged a career as a pop singer, but found that she was too shy for life in the spotlight and "hated being on stage".
As she accepted her prize, she praised her education at Putteridge High School, a comprehensive in Luton, and criticised the government's attitude towards the arts.
Jude Law, who had presented Price with her award, had earlier criticised the "cultural vandalism" of the new English Baccalaureate qualification, which he said would take "arts, design and music" out of schools.
Price said "It's incredibly depressing listening to the comments people made earlier that a young girl from Luton going to a comprehensive might not be able to imagine being an artist and might not have the opportunities I've had."
Elizabeth Price's Turner Prize-winning artwork includes a video inspired by a devastating fire in Manchester in the 1970s.
"The Woolworths Choir Of 1979" features archive footage from the blaze, which gutted a city-centre store and left 10 people dead. That material is mixed with music, images of architectural designs and film clips of a 1960s girl group.
Watch a clip from the video installation below.