We present a novel algorithm called Dynamic Perturbation for solving large-scale macroeconomic models. Our approach involves computing first-order Taylor expansions of the policy functions along the entire equilibrium path. This method applies to a wide range of models and offers significantly higher accuracy than traditional perturbation approaches. Remarkably, even when utilising first-order approximations, our method can effectively handle models with strong nonlinearities and occasionally binding constraints, such as the zero lower bound.