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.