The dynamics of the core of the Sagittarius (Sgr) dwarf spheroidal (dSph) galaxy are explored using high-resolution (R ~ 22, 500), H-band, near-infrared spectra of over 1000 giant stars in the central 3 deg2 of the system, of which 328 are identified as Sgr members. These data, among some of the earliest observations from the Sloan Digital Sky Survey III/Apache Point Observatory Galactic Evolution Experiment (APOGEE) and the largest published sample of high resolution Sgr dSph spectra to date, reveal a distinct gradient in the velocity dispersion of Sgr from 11 to 14 km s–1 for radii >0.°8 from center to a dynamical cold point of 8 km s–1 in the Sgr center—a trend differing from that found in previous kinematical analyses of Sgr over larger scales that suggests a more or less flat dispersion profile at these radii. Well-fitting mass models with either cored and cusped dark matter distributions can be found to match the kinematical results, although the cored profile succeeds with significantly more isotropic stellar orbits than required for a cusped profile. It is unlikely that the cold point reflects an unusual mass distribution. The dispersion gradient may arise from variations in the mixture of populations with distinct kinematics within the dSph; this explanation is suggested (e.g., by detection of a metallicity gradient across similar radii), but not confirmed, by the present data. Despite these remaining uncertainties about their interpretation, these early test data (including some from instrument commissioning) demonstrate APOGEE's usefulness for precision dynamical studies, even for fields observed at extreme airmasses.