Measurement in power systems and, particularly, in smart grids and smart microgrids is often concerned with harmonic analysis of voltage and current waveforms. The use of Fourier-based algorithms is widespread, and the limits following from the fundamental time-versus-frequency tradeoff that relates observation time to frequency resolution are well understood. This paper presents the application of an algorithm based on the principles of compressive sensing that can achieve an order-of-magnitude resolution improvement without significantly extending total observation time. For harmonic analysis in power systems, this means that accurate results can be obtained using shorter observation intervals, which allow to effectively track changes and reduce the effect of transients on measurements. The application of the algorithm to harmonics and interharmonics, as well as to phasor measurement, is considered and analyzed.