We study the spatiotemporal dynamics of random spatially distributed noninfinitesimal perturbations in one-dimensional chaotic extended systems. We find that an initial perturbation of finite size epsilon0 grows in time obeying the tangent space dynamic equations (Lyapunov vectors) up to a characteristic time tx(epsilon0) approximately b-(1/lambda(max))ln(epsilon0), where lambda(max) is the largest Lyapunov exponent and b is a constant. For times t<tx, perturbations exhibit spatial correlations up to a typical distance xi approximately tz. For times larger than tx, finite perturbations are no longer described by tangent space equations, memory of spatial correlations is progressively destroyed, and perturbations become spatiotemporal white noise. We are able to explain these results by mapping the problem to the Kardar-Parisi-Zhang universality class of surface growth.