We present 2.0-2.4 mu m integral field spectroscopy at adaptive optics spatial resolution (similar to 0.1 '') obtained with the Near-infrared Integral Field Spectrograph (NIFS) at Gemini North Observatory of six classical T Tauri stars: T Tau, DG Tau, XZ Tau, HL Tau, RW Aur, and HV Tau C. In all cases, the v = 1-0 S(1) (2.12 mu m) emission is detected at spatially extended distances from the central stars. Moreover, HL Tau, T Tau, RW Aur, and HV Tau C have H-2 that extends to projected distances of more than similar to 200 AU from the stars. Integrated over the IFU field, most of the H-2 emission is not spatially coincident with the location of continuum flux. Multiple H-2 transitions detected in the K-band spectra show that level populations are typical of gas in thermal equilibrium with excitation temperatures in the 1800-2300 K range. Three of the stars have H-2 velocity profiles that are centered approximately at the stellar radial velocity, and three show velocity shifts with respect to the system. Each of the stars studied here exhibit H-2 morphologies, spatial extents, excitation temperatures, and kinematics that are most consistent with shock-excited emission from the inner regions of the known Herbig-Haro energy flows or from spatially extended wide-angle winds encompassing the outflows rather than predominantly from H-2 stimulated quiescently by UV or X-ray emission from the central stars. The data presented in this study highlights the sensitivity of adaptive-optics-fed integral field spectroscopy for spatially resolving emission line structures in the environments of bright young stars.