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We propose a novel algorithm for sparse system identification in the frequency domain. Key to our result is the observation that the Fourier transform of the sparse impulse response is a simple sum of complex exponentials, whose parameters can be efficiently determined from only a narrow frequency band. From this perspective, we present a sub-Nyquist sampling scheme, and show that the original continuous-time system can be learned by considering an equivalent low-rate discrete system. The impulse response of that discrete system can then be adaptively obtained by a novel frequency-domain LMS filter, which exploits the parametric structure of the model. Numerical experiments confirm the effectiveness of the proposed scheme for sparse system identification tasks.