Quasiperiodic radio frequency interference (RFI), such as those generated by telecommunication and active radar systems, is commonly encountered in radio astronomy observations. Such RFI-contaminated signals contain hidden periodicities due to cyclic features involved in their formation (e.g., carrier frequencies, periodic keying of the amplitude, and baud rates). RFI signal characterization and its subsequent excision based on the well-known cyclic spectrum analysis have been previously demonstrated; however, the high complexity of the algorithm and the computational cost of its implementation have limited its utility in radio astronomy, rendering less sophisticated solutions. To overcome this challenge, we present a novel method for RFI detection and mitigation based on efficient estimation of the cyclic spectrum by compressive statistical sensing (CSS) of sub-Nyquist data. CSS performs second-order statistical estimation such as cyclic spectrum using a reduced number of input samples, thereby enabling accelerated performance. To validate the feasibility of the proposed method, we conduct experiments with simulated data and assess the detection and mitigation results under different parameter settings, for example, interference-to-noise ratio, additional RFI sources, frequency resolution, and input data size. We demonstrate the real performance of the method by analyzing radio astronomy data (∼1.3 GHz) acquired with the L-wide band receiver at the Arecibo Observatory, which is typically corrupted by active air surveillance radars located nearby. Our CSS-based solution enables robust and efficient detection of the RFI frequency bands present in the L-band data, and subsequent excision by blanking is also demonstrated.