We demonstrate the existence of a simple scaling form which describes the crossover from anomalous to Gaussian exponents as a function of cutoff in the Zhang model of surface growth for μ> 2, where μ is the exponent characterising the power-law noise distribution. For μ= 2 we find a novel scaling form with logarithmic corrections. Our results for cutoff scaling at μ= 2 are supported by the scaling of the saturation growth velocity.