Growing observational evidence has suggested active meteorology in the atmospheres of brown dwarfs (BDs) and directly imaged extrasolar giant planets (EGPs). In particular, a number of surveys have shown that near-IR brightness variability is common among L and T dwarfs. Despite the likelihood from previous studies that atmospheric dynamics is the major cause of the variability, the detailed mechanism of the variability remains elusive, and we need to seek a natural, self-consistent mechanism. Clouds are important in shaping the thermal structure and spectral properties of these atmospheres via their opacity, and we expect the same for inducing atmospheric variability. In this work, using a time-dependent one-dimensional model that incorporates a self-consistent coupling between the thermal structure, convective mixing, cloud radiative heating/cooling and condensation/evaporation of clouds, we show that radiative cloud feedback can drive spontaneous atmospheric variability in both temperature and cloud structure under conditions appropriate for BDs and directly imaged EGPs. The typical periods of variability are one to tens of hours with typical amplitude of the variability up to hundreds of Kelvins in effective temperature. The existence of variability is robust over a wide range of parameter space, but the detailed evolution of the variability is sensitive to model parameters. Our novel, self-consistent mechanism has important implications for the observed flux variability of BDs and directly imaged EGPs, especially for objects whose variability evolves on short timescales. It is also a promising mechanism for cloud breaking, which has been proposed to explain the L/T transition of brown dwarfs. Subject headings: brown dwarfs -planets and satellites: gaseous planets -planets and satellites:atmospheres -radiative transfer -methods: numerical arXiv:1809.06467v2 [astro-ph.SR]