We present the abundance analysis for a sample of 7 red giant branch stars in the metal-poor globular cluster NGC 4372 based on UVES spectra acquired as part of the Gaia-ESO Survey. This is the first extensive study of this cluster from high resolution spectroscopy. We derive abundances of O, Na, Mg, Al, Si, Ca, Sc, Ti, Fe, Cr, Ni, Y, Ba, and La. We find a metallicity of [Fe/H] = -2.19 $\pm$ 0.03 and find no evidence for a metallicity spread. This metallicity makes NGC 4372 one of the most metal-poor galactic globular clusters. We also find an {\alpha}-enhancement typical of halo globular clusters at this metallicity. Significant spreads are observed in the abundances of light elements. In particular we find a Na-O anti-correlation. Abundances of O are relatively high compared with other globular clusters. This could indicate that NGC 4372 was formed in an environment with high O for its metallicity. A Mg-Al spread is also present which spans a range of more than 0.5 dex in Al abundances. Na is correlated with Al and Mg abundances at a lower significance level. This pattern suggests that the Mg-Al burning cycle is active. This behavior can also be seen in giant stars of other massive, metal-poor clusters. A relation between light and heavy s-process elements has been identified.