We propose a new computational method for the valuation of options in jump-diffusion models. The option value function for European and barrier options satisfies a partial integro-differential equation (PIDE). This PIDE is commonly integrated in time by implicit-explicit (IMEX) time discretization schemes, where the differential (diffusion) term is treated implicitly, while the integral (jump) term is treated explicitly. In particular, the popular IMEX Euler scheme is first order accurate in time. Second order accuracy in time can be achieved by using the IMEX midpoint scheme. In contrast to the above approaches, we propose a new high-order time discretization scheme for the PIDE based on the extrapolation approach to the solution of ODEs, that also treats the diffusion term implicitly and the jump term explicitly. The scheme is simple to implement, can be added to any PIDE solver based on the IMEX Euler scheme, and is remarkably fast and accurate. We demonstrate our approach on the examples of Merton's and Kou's jump-diffusion models, diffusion-extended Variance Gamma model, as well as the two-dimensional Duffie-Pan-Singleton model with correlated and contemporaneous jumps in the stock price and its volatility. By way of example, pricing a one-year double-barrier option in Kou's jump-diffusion model, our scheme attains accuracy of $10^{-5}$ in 72 time steps (in 0.05 seconds). In contrast, it takes the first-order IMEX Euler scheme more than 1.3 million time steps (in 873 seconds) and the second-order IMEX midpoint scheme 768 time steps (in 0.49 seconds) to attain the same accuracy. Our scheme is also well suited for Bermudan options. Combining simplicity of implementation and remarkable gains in computational efficiency, we expect this method to be very attractive to financial engineering modelers.