The behavior of interacting spins subject to randomness is a longstanding issue and the emergence of exotic quantum states is among intriguing theoretical predictions. We show how a quantum-disordered phase emerges from a classical antiferromagnet by controlled randomness. ^{1}H NMR of a successively x-ray-irradiated organic Mott insulator finds that the magnetic order collapses into a spin-glass-like state, immediately after a slight amount of disorder centers are created, and evolves to a gapless quantum-disordered state without spin freezing, spin gap, or critical slowing down, as reported by T. Furukawa et al. [Phys. Rev. Lett. 115, 077001 (2015)]PRLTAO0031-900710.1103/PhysRevLett.115.077001 through sequential reductions in the spin freezing temperature and moment.