Millisecond pulsars are very likely the main source of gamma-ray emission from globular clusters. However, the relative contributions of two separate emission processes–curvature radiation from millisecond pulsar magnetospheres vs. inverse Compton emission from relativistic pairs launched into the globular cluster environment by millisecond pulsars–have long been unclear. To address this, we search for evidence of inverse Compton emission in 8-yearFermi-LAT data from the directions of 157 Milky Way globular clusters. We find that the gamma-ray luminosities of the globular clusters are correlated with their stellar encounter rates (6.4σ) and total radiation field energy density (3.8σ). We demonstrate that the gamma-ray emission of globular clusters can be resolved spectrally into two components: i) an exponentially cut-off power law and ii) a pure power law. The latter component–which we uncover at a significance of 8.2σ–is most naturally interpreted as inverse Compton emission by cosmic-ray electrons and positrons injected by millisecond pulsars. The luminosity of this inverse Compton component suggests that the fraction of millisecond pulsar spin-down luminosity into relativistic leptons is similar to the fraction of the spin-down luminosity into prompt magnetospheric radiation.