Winning the Turner prize can take a person in all kinds of ways: artists can be bashful, weepy, overcome, euphoric, or any variation thereof. On Monday night, when 46-year-old Elizabeth Price was handed the cheque, it was a notably composed woman who gave her acceptance speech: she thanked her fellow shortlisted artists; paid delicate tribute to Michael Stanley, the Turner prize judge and director of Modern Art Oxford who died this autumn; and, finally, launched into an impassioned critique of the Ebacc, the proposed certificate at 16 for school pupils who have passed GCSEs in five subject areas – the arts not among them.  When we speak the following morning the artist, who won the prize for her powerful, allusive video work, is equally eloquent on what she regards as the urgent politics of the arts. She says of the previous evening's moment on the podium, "I felt very grave and unelated, and serious, and focused." Her career, she says, would have been impossible without the "generous opportunities I've had through education and public funding". She sees a triple blow: the introduction of the Ebacc; the removal of state funding for the arts in universities; and the reduction of the arts budget – with more cuts feared. "What's depressing about the Ebacc is not only that it will be difficult for individuals to fulfil their ambitions, or get to identify their capabilities and shape their lives around them. But also what you end up with is art becoming something that is available only to privileged people, and expressive only of that experience. That's also what's so damaging about the withdrawal of state funding for humanities and arts at universities: these will become the subjects of the privileged, and history-writing and novel-writing and art-making and poetry-writing will become homogeneous."  Price is not one of those artists who has made a fortune, or even much of a living, from her work. She teaches a couple of days a week ("that's how I pay the bills"), and for many years "scrapped around" for jobs. It wasn't until she was in her 40s that her career took off. Even five years ago, she says, "the idea that I would be nominated for the Turner prize was just absurd, preposterous. Completely off the map."  Until then, she had been working in a tradition of pop-inflected conceptual art. But there was something missing. The work felt "too polite, too nervous". She remembers looking at it and thinking: "It's not like I am. It's not pissed off about things. The jokes aren't the jokes I find funny. It's too self-conscious and not direct enough and not candid enough." She considered dropping the whole thing, giving up art: she would mentally scroll through alternative careers, like social work. Then she started making video, and with the first piece, called A Public Lecture and Exhumation, which combined a sort of documentary-style text with images, she felt completely "emancipated". The work ventriloquised the kind of language used by local-authority committees – a language she knew very well: for six years she worked as a part-time administrator for Hackney Council in London. She was a pretty bad administrator, she says, but she learned a lot.  The work on display now at Tate Britain, The Woolworths Choir of 1979, takes three apparently disparate elements: church architecture; the 1960s girl band the Shangri-Las; and the fire that killed 10 people in the central Manchester branch of Woolworths. The choir of a church is a place of assembly and singing, which takes us to the 1960s pop music; and all three elements are connected by a curious twisted-arm gesture – a girl's arm waving from the burning store; the choreographed dance movements of the pop singers; the arms of figures on medieval funerary monuments. An apparently authoritative, documentary-style text accompanies the images. The piece's power is impossible to articulate in writing, but there is something in its poetic confidence that is completely compelling.  Not bad, in fact, for someone who did "disastrously" in her first year at art school, and "failed all my exams". That was at the Ruskin School of Drawing at Oxford University in the mid-1980s, where she now teaches. At first, she says, she was lonely and "a bit chippy". She came without the ready-made social circle that she saw those from grander schools bring; she had been to a Luton comprehensive, which she praises. "Anyone could be in the orchestra, or sports team, or arts club at my school. It was precisely the kind of inclusivity that now meets with a sort of scorn and derision as a prizes-for-all culture that generates only mediocrity. There's something so insulting about the idea that including lots of people means mediocrity." She met Amelia Fletcher, now a prominent economist: the two would go on to form the successful indie band Talulah Gosh. Price left the band in the end – she hated being on stage, she said. But the group meant she had "friends and a social life"; and then her art and everything else went better, too.  Working in video means she can finally bring music back into her work. "I have a pretty expanded view of what art is. I include pop music and even some sports," she says. What sports, I wonder? "Well – football and cricket," she says. "Part of what makes them enjoyable is the culture, the language. I used to listen to the cricket in my studio. I have never played cricket, I have no idea what the rules are, but I enjoyed the terminology."  There have been 28 winners of the Turner prize, but Price is only the fifth woman. Why? "Because it's difficult to have a career as an artist, and in every situation where it's difficult to have a career, it's even harder for women, for all the other reasons that it's harder in other fiercely contested fields. For women who have children, the economic difficulty of sustaining a life as an artist maybe makes it impossible. There's no maternity leave, there's no pension." Price has no children, but she does have a big family of proud siblings who gathered at Monday's prize ceremony. The award has, she says, involved her not-especially-arty parents in "this slightly cranky thing that I did" – an experience that has been "completely unexpected and very pleasurable".  In the end, though, we are back to politics. Price is hugely concerned about the government's position on the arts: "It doesn't even look very thought-through. It looks like an ideologically driven set of prejudices." What she is also clear on, is that her career – moulded as it has been by a comprehensive and free university education, and supported by public funding for the arts – "would be impossible now. And the Ebacc is part of that."
She failed all her art-school exams, joined a band and didn't get going until she was in her 40s. Now Elizabeth Price has won the Turner prize. She talks to Charlotte Higgins about the long game