We propose a fast algorithm to reconstruct spectrally sparse signals from a small number of randomly observed time domain samples. Different from conventional compressed sensing where frequencies are discretized, we consider the super-resolution case where the frequencies can be any values in the normalized continuous frequency domain [0; 1). We first convert our signal recovery problem into a low rank Hankel matrix completion problem, for which we then propose an efficient feasible point algorithm named projected Wirtinger gradient algorithm(PWGA). The algorithm can be further accelerated by a scheme inspired by the fast iterative shrinkage-thresholding algorithm (FISTA). Numerical experiments are provided to illustrate the effectiveness of our proposed algorithm. Different from earlier approaches, our algorithm can solve problems of large scale efficiently.