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 versus 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-yr Fermi-LAT data from the directions of 157 Milky Way globular clusters. We find a mildly statistically significant (3.8{sigma}) correlation between the measured globular cluster gamma-ray luminosities and their photon field energy densities. However, this may also be explained by a hidden correlation between the photon field densities and the stellar encounter rates of globular clusters. Analysed in toto, 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 8.2{sigma} - has a power index of 2.79 +/- 0.25. It is most naturally interpreted as inverse Compton emission by cosmic-ray electrons and positrons injected by millisecond pulsars. We find the luminosity of this power-law component is comparable to, or slightly smaller than, the luminosity of the curved component, suggesting 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.