High-resolution H-band spectra of five bright field K, M, and MS giants, obtained from the archives of the Kitt Peak National Observatory Fourier transform spectrometer, are analyzed to determine chemical abundances of 16 elements. The abundances were derived via spectrum synthesis using the detailed linelist prepared for the Sloan Digital Sky Survey III Apache Point Galactic Evolution Experiment (APOGEE), which is a high-resolution near-infrared spectroscopic survey to derive detailed chemical abundance distributions and precise radial velocities for 100,000 red giants sampling all Galactic stellar populations. The red giant sample studied here was chosen to probe which chemical elements can be derived reliably from the H-band APOGEE spectral region. These red giants consist of two K-giants (α Boo and μ Leo), two M-giants (β And and δ Oph), and one thermally pulsing asymptotic giant branch (TP-AGB) star of spectral type MS (HD 199799). Measured chemical abundances include the cosmochemically important isotopes 12C, 13C, 14N, and 16O, along with Mg, Al, Si, K, Ca, Ti, V, Cr, Mn, Fe, Co, Ni, and Cu. The K and M giants exhibit the abundance signature of the first dredge-up of CN-cycle material, while the TP-AGB star shows clear evidence of the addition of 12C synthesized during 4He-burning thermal pulses and subsequent third dredge-up. A comparison of the abundances derived here with published values for these stars reveals consistent results to ~0.1 dex. The APOGEE spectral region and linelist is thus well suited for probing both Galactic chemical evolution, as well as internal nucleosynthesis and mixing in populations of red giants via high-resolution spectroscopy.