I can't recall any novel, classified as Science Fiction/Fantasy by author/general public, that has won any of these awards. Are there any?
That's difficult to answer, because the SF-ness of a work is not an objective criterion. There's the added difficulty that the Nobel prize is usually awarded to an author for the sum of his work, not to a specific work.
Taking having an entry on ISFDB as a criterion, here's a list (not exhaustive, I just tried names that I thought might be considered ISFDB material). They tend not to be published as SF — though some came too early and are now considered classics, and some would probably be published as SF if they weren't from a non-habitually-SF writer. Several are typically classified as magical realism — when supernatural elements are used in a story but play a background, decidedly metaphorical role.
- Rudyard Kipling, Nobel 1907, well-known for stories involving talking animals (most famously The Jungle Book), but these are generally considered children's literature.
- Selma Lagerlöf, Nobel 1909, now mainly remembered for The Wonderful Adventures of Nils, a children's book involving a child transformed into a pixie, but that book may not have played a big part in her being awarded the prize.
- William Faulkner, Nobel 1949, Pulitzer 1955 for A Fable and 1963 for The Reivers. A Fable is sometimes classified as speculative fiction.
- Isaac Bashevis Singer, Nobel 1978, sometimes incorporated supernatural elements from Jewish folklore in his tales.
- Gabriel Garcia Marquez, Nobel 1982, played on the passage of time in One Hundred Years of Solitude.
- William Golding, Nobel 1983, wrote some stories with SF-nal elements, such as The Inheritors.
- Günter Grass, Nobel 1999, occasionally used SF-nal elements in his novels, most famously The Tin Drum (whose protagonist decided to stop growing up at age three).
- Doris Lessing, Nobel 2007, wrote a few stories that can be considered SF.
- Midnight's Children by Salman Rushdie, Booker 1981, uses an SF-nal element (telepathy).
- The Blind Assassin by Margaret Atwood, Booker 2000, has a science-fiction story within the story, which turns out to be closely linked to the main story.
- The Famished Road by Ben Okri, incorporates supernatural elements from Nigerian folklore.
- The Amazing Adventures of Kavalier & Clay by Michael Chabon, Pulitzer 2001, is technically not SF, but full of SF-nal elements.
- The Road by Cormac McCarthy, Pulitzer 2007, is post-apocalyptic and thus firmly SF.
None of the authors in this list are generally considered SF writers, though Michael Chabon comes close.
The Road by Cormac McCarthy won the Pulitzer Prize a few years back.
The Nobel Prize for Literature is not awarded on the basis of a single novel but is given to an author for their contribution to the world of literature. A quick scan of the list of Laureates suggests Doris Lessing who wrote the Canopus in Argos sequence which is most definitely Science Fiction.
The Booker prize list didn't suggest any Science Fiction although The Famished Road by Ben Okri has some fantastic elements (the Spirit World is perceived by the main character).
Harry Martinson was a Nobel laureate. One of his best-known works is Aniara, about a starship. Strictly speaking, Aniara isn't a novel, it's an epic poem. But it's book-length, and it's science-fictional enough that we carried it (along with many of the titles in Gilles' awesome list) at an SF bookstore I managed in the late 80s.
Add Winston Churchill to the list -- Nobel Prize for literature, and he wrote some alternate history SF ("What If Lee Had Not Won The Battle Of Gettysburg").
Hermann Hesse, Nobel 1946 for The Glass Bead Game aka Magister Ludi -- a work set about four centuries from now, centering on a game of intellect. Among the people it has inspired are the architect Christopher Alexander, whose Pattern Language is arguably a form of the Game, and the "father" of genetic algorithms, John Holland.
Speculative fiction should be proud to claim it.