Well, not consciousness, per se. Let's call it "literary consciousness." Or "narrative consciousness."
It's not that narrators before Faulkner were unconscious. It's that they were too conscious.
It all started because Faulkner gave up. After his first two books (fronted by the novelist Sherwood Anderson) went nowhere and his third was roundly rejected by publishers (and finally brought out in a highly edited form), he threw caution to the wind and created two novels back to back, THE SOUND AND THE FURY and AS I LAY DYING, that reshaped how literary characters went about the business of being characters. He sparked a change that we now take for granted.
In these two novels, the characters--speakers, all--are not rational. Instead, we hear a stream of their confessions and dreams, sieved through their unconscious. It's almost as if they're talking in their sleep, revealing deeply personal things about themselves in haphazard and sometimes misleading ways (ahem, Dewey Dell). That is, about the way your own unconscious self operates.
When Esther steps forward to tell her tale in Dickens' BLEAK HOUSE or when Moll Flanders narrates her life in Defoe's novel of the same name, they tell their tales in a rational, cohesive way. Start here at point A, move to point B, tell why A led to B, go on to C, and so on. They offer linear movement through their lives.
The characters in AS I LAY DYING offer something else. Call it, not narrative, but "performance." It's the self turned inside out, a messy jumble. Cora is trying to hold it together with religion. Darl is slowly going mad. Vardaman can only understand the world as a small child. Cash and Jewel are blinded by their dreams and regrets.
The only one who gives us any sort of linearity is the unseen narrator who is stitching it all together, calling the performers on stage one by one to make their pitch.
This is Faulkner: he narrated the unconscious. He tapped something almost untellable. He found below his characters the wellsprings, not of themselves, but of their imaginations (that is, of who they think they are).
His narrative technique opened the door to Philip Roth. You can't have Portnoy without Faulkner! It opened the door to Saul Bellow, to Flannery O'Connor, to Louise Erdrich (many consider her Faulkner's direct heir), to Toni Morrison (another direct heir), to Don Delillo, to Salmon Rushdie (yet another direct heir), to Ian McEwan, even to Alan Bennett.
In the end, I would posit that Faulkner has become more influential on literary writers than almost any other English-speaking author of the twentieth century. James Joyce was undoubtedly smarter, but his voice always stays in the head, a chilly intellectualism. Faulkner's voice is located further down and farther in, somewhere in the body.