We show that the defect density n, for a slow nonlinear power-law quench with a rate tau(-1) and an exponent alpha > 0, which takes the system through a critical point characterized by correlation length and dynamical critical exponents nu and z, scales as n similar to tau(-alpha nu d/(alpha z nu+1)) [n similar to(alpha g((alpha-1)/alpha)/tau)(nu d/(z nu+1))] if the quench takes the system across the critical point at time t=0 [t=t(0)not equal 0], where g is a nonuniversal constant and d is the system dimension. These scaling laws constitute the first theoretical results for defect production in nonlinear quenches across quantum critical points and reproduce their well-known counterpart for a linear quench (alpha=1) as a special case. We supplement our results with numerical studies of well-known models and suggest experiments to test our theory.