Well, officially, Chuck Yeager in the Bell X-1 in 1947. Did someone beat him to it?
In this enjoyable Skeptoid article, Brian Dunning breaks down the possibilities, all concerning claims made for planes in power dives (Yeager did it in level flight, and there has been no argument that he was the first to do that).
One idea is that a German Me-163 rocket fighter did it during WWII, but Dunning rejects this common Internet-posted claim. He argues the Komet, one of the earliest delta-wing planes, could not have done it: the design and the control surfaces were such that the plane could have hit the sound barrier only in an uncontrolled dive ending in death, not in a recoverable maneuver as claimed only in a 1990 book, not verified in any captured German test records.
An intriguing report says Hans Guido Mutke, a German Me-262 jet fighter pilot, believed he'd passed the sound barrier only in retrospect, after he'd read of Yeager's experiences and the phenomena he'd encountered passing through the barrier. This falls into the "highly unlikely but not impossible" bin. The Me-262 was not designed for supersonic flight and would likely have torn apart or been destroyed by the phenomenon known as "Mach tuck," in which the nose is forced downward until the pilot loses control.
American pilot George Welch believes he broke the barrier twice before Yeager's flight by diving steeply in the XP-86 prototype fighter plane. Since Welch did the same thing again, properly instrumented and verified, in the same aircraft AFTER Yeager's record was set, it seems likely he did it before Chuck as well. Likely - but not proven or provable in the official sense.
None of this detracts from the quantum leap in capability pioneered by the X-1 and Chuck Yeager. He still holds the official record and likely always will.