Just to clarify: I am explicitly not referring to the Bell X-1, or any other early experimental supersonic aircraft that were rocket-powered. I'm curious as to the first supersonic aircraft powered by an air-breathing jet engine, as well as (if possible) the technological innovations needed to transition from rocket-powered to air-breathing supersonic flight.
One ME-262 Pilot claims to have exceeded Mach 1 in a dive, but the conditions are disputed due to compressibility and later tests that find the aircraft uncontrollable above Mach .84. The first U.S. fighter plane to exceed Mach 1 in level flight is the F-100 Super Sabre.
For research on how the sound barrier was broken as it results to aircraft design is to read Wikipedia: Sound Barrier.
Wikipedia also has a good read on Supersonic Aircraft that lists aircraft by manufacture date.
A member of the Century Series, the F-102 was the USAF's first operational supersonic interceptor and delta-wing fighter. (Wikipedia: Convair F-102 Delta Dagger)
Its innovative "coke bottle" fuselage design to lower the drag at the wing root was critical in achieving supersonic flight.
While not capable of level-flight supersonic performance, the Korean War era F-86 Sabrejet was capable of slightly exceeding Mach 1 (I recall the figure as Mach 1.05) in a shallow dive. This was one of its performance points over the (otherwise similar) MiG-15, which was Mach-limited and became nearly uncontrollable in the transonic regime.
While not generally considered "supersonic" the F-86 was capable of limited supersonic flight, and had several important features that were common to later truly supersonic designs -- sharply swept wings and fuselage-mounted drag brake panels, for example.