We present spectroscopic determinations of the effective temperatures, surface gravities, and metallicities for 21 M dwarfs observed at high resolution (R ∼ 22,500) in the H band as part of the Sloan Digital Sky Survey (SDSS)-IV Apache Point Observatory Galactic Evolution Experiment (APOGEE) survey. The atmospheric parameters and metallicities are derived from spectral syntheses with 1D LTE plane-parallel MARCS models and the APOGEE atomic/molecular line list, together with up-to-date H2O and FeH molecular line lists. Our sample range in Teff from ∼3200 to 3800 K, where 11 stars are in binary systems with a warmer (FGK) primary, while the other 10 M dwarfs have interferometric radii in the literature. We define an ${M}_{{K}_{S}}$ M K S ─radius calibration based on our M-dwarf radii derived from the detailed analysis of APOGEE spectra and Gaia DR2 distances, as well as a mass─radius relation using the spectroscopically derived surface gravities. A comparison of the derived radii with interferometric values from the literature finds that the spectroscopic radii are slightly offset toward smaller values, with ∆ = −0.01 ± 0.02 R⋆/R☉. In addition, the derived M-dwarf masses based upon the radii and surface gravities tend to be slightly smaller (by ∼5%─10%) than masses derived for M-dwarf members of eclipsing binary systems for a given stellar radius. The metallicities derived for the 11 M dwarfs in binary systems, compared to metallicities obtained for their hotter FGK main-sequence primary stars from the literature, show excellent agreement, with a mean difference of [Fe/H](M dwarf ─ FGK primary) = +0.04 ± 0.18 dex, confirming the APOGEE metallicity scale derived here for M dwarfs.