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We report the first detailed chemical abundances for five globular clusters (GCs) in M31 from high-resolution (R ∼ 25,000) spectroscopy of their integrated light (IL). 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 GC population. The data presented here were obtained with the HIRES echelle spectrograph on the Keck I telescope and are analyzed using a new IL 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 Gyr, 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 α-elements Ca, Si, and Ti relative to Fe. We also find [Mg/Fe] to be low relative to other [α/Fe], and [Al/Fe] to be enhanced in the IL abundances. These results imply that abundances of Mg, Al (and likely O, Na) recovered from IL 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 IL abundances of GCs 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 five M31 GCs, as well as independent constraints on the reddening toward the clusters from our analysis.