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.
DH108 (de Havilland) Powered by the 3000lb thrust DH Goblin and piloted by John Derry 6th Sept 1948 achieved Mach 1.07 in a dive.
According to an account published in "The Evening Star", Washington, DC (Sep 10, 1948), p. A-19:
In 1948, the Royal Aero Club awarded John Derry the Gold Medal for being the first British subject to exceed the speed of sound.
For more information, see Brian Rivas and Annie Bullen, "John Derry: The Story of Britain's First Supersonic Pilot" (1982)