With good electronic counter-countermeasures capability, random pulse repetition interval (PRI) radars receive increasing attention recently. However, the problem of maneuvering target detection in random PRI radars is rarely studied yet. In this problem, the difficulty lies in not only the range migration (RM) and Doppler frequency migration (DFM) effects but also the non-uniform sampling pulses. This paper proposes a novel algorithm for this problem. In the proposed algorithm, we first combine the non-uniform resampling operation and the keystone transform to propose the resampling-keystone transform, which can eliminate the RM and resample the non-uniform sampling pulses into uniform ones in one step. Then, the dechirp process, whose implementation can benefit from the fast Fourier transform, is employed to accomplish coherent integration for target detection by compensating the DFM. The proposed algorithm is applicable for both single-target and multi-target scenarios. Besides, the available integration time and computational complexity of the proposed algorithm are analyzed. Finally, simulation results are given to show that the proposed algorithm can approach the optimal detection performance with a much lower computational cost than the well-known generalized Radon Fourier transform.