In this paper I develop a new computational method for pricing path-dependent options. Using the path integral representation of the option price, I show that in general it is possible to perform analytically a partial averaging over the underlying risk-neutral diffusion process. This result greatly eases the computational burden placed on the subsequent numerical evaluation. For short- to medium-term options it leads to a general approximation formula that requires only the evaluation of a one-dimensional integral. I illustrate the application of the method to Asian options and occupation-time derivatives.