We report on Nuclear Spectroscopic Telescope Array (NuSTAR), Neil Gehrels Swift Observatory (Swift) X-ray Telescope (XRT), and Swift Burst Alert Telescope (BAT) observations of IGR J16493-4348, a wind-fed supergiant X-ray binary showing significant superorbital variability. From a discrete Fourier transform of the BAT light curve, we refine its superorbital period to be 20.058 ± 0.007 days. The BAT dynamic power spectrum and a fractional root mean square analysis both show strong variations in the amplitude of the superorbital modulation, but no observed changes in the period are found. The superorbital modulation is significantly weaker between MJD 55,700 and MJD 56,300. The joint NuSTAR and XRT observations, which were performed near the minimum and maximum of one cycle of the 20 day superorbital modulation, show that the flux increases by more than a factor of two between superorbital minimum and maximum. We find no significant changes in the 3-50 keV pulse profiles between superorbital minimum and maximum, which suggests a similar accretion regime. Modeling the pulse-phase-averaged spectra we find a possible Fe Kα emission line at 6.4 keV at superorbital maximum. This feature is not significant at superorbital minimum. While we do not observe any significant differences between the pulse-phase-averaged spectral continua apart from the overall flux change, we find that the hardness ratio near the broad main peak of the pulse profile increases from superorbital minimum to maximum. This suggests the spectral shape hardens with increasing luminosity. We discuss different mechanisms that might drive the observed superorbital modulation.