We report the first detailed chemical abundances for 5 globular clusters (GCs) in M31 from high-resolution (R ~ 25,000) spectroscopy of their integrated light. These GCs are the first in a larger set of clusters observed as part of an ongoing project to study the formation history of M31 and its globular cluster population. The data presented here were obtained with the HIRES echelle spectrograph on the Keck I telescope, and are analyzed using a new integrated light spectra analysis method that we have developed. In these clusters, we measure abundances for Mg, Al, Si, Ca, Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Y, and Ba, ages >10 Gyrs, and a range in [Fe/H] of -0.9 to -2.2. As is typical of Milky Way GCs, we find these M31 GCs to be enhanced in the alpha-elements Ca, Si, and Ti relative to Fe. We also find [Mg/Fe] to be low relative to other [alpha/Fe], and [Al/Fe] to be enhanced in the integrated light abundances. These results imply that abundances of Mg, Al (and likely O, Na) recovered from integrated light do display the inter- and intra-cluster abundance variations seen in individual Milky Way GC stars, and that special care should be taken in the future in interpreting low or high resolution integrated light abundances of globular clusters that are based on Mg-dominated absorption features. Fe-peak and the neutron-capture elements Ba and Y also follow Milky Way abundance trends. We also present high-precision velocity dispersion measurements for all 5 M31 GCs, as well as independent constraints on the reddening toward the clusters from our analysis.