The conventional algorithms for parameter estimation of the cubic phase signal cannot achieve a trade-off between the low computational cost and high anti-noise performance. Considering the question, a novel method based on the nonuniform fast Fourier transform (NUFFT) is proposed to estimate parameters of multi-component cubic phase signal. Firstly, the improved HAF-ICPF method based on NUFFT is introduced. Then, the improved HAF-ICPF method is utilized to finish the quadratic chirp rate estimation of multi-component cubic phase signal. The proposed method has low computational complexity due to avoiding the matching process of searching and still has same performance with the HAF-ICPF method. Simulations verify the effectiveness of the method.