We present Nuclear Spectroscopic Telescope Array (NuSTAR) hard X-ray timing and spectroscopy of the three exceptionally energetic rotation-powered millisecond pulsars PSRs B1821-24, B1937+21, and J0218+4232. By correcting for the frequency and phase drifts of the NuSTAR onboard clock, we are able to recover the intrinsic hard X-ray pulse profiles of all three pulsars with a resolution down to . The substantial reduction of background emission relative to previous broadband X-ray observations allows us to detect for the first time pulsed emission up to ∼50 keV, ∼20 keV, and ∼25 keV for the three pulsars, respectively. We conduct phase-resolved spectroscopy in the 0.5-79 keV range for all three objects, obtaining the best measurements yet of the broadband spectral shape and high-energy pulsed emission to date. We find extensions of the same power-law continua seen at lower energies, with no conclusive evidence for a spectral turnover or break. Extrapolation of the X-ray power-law spectrum to higher energies reveals that a turnover in the 100 keV to 100 MeV range is required to accommodate the high-energy γ-ray emission observed with Fermi-LAT, similar to the spectral energy distribution observed for the Crab pulsar.