The Gaia-ESO survey is a large public spectroscopic survey aimed at investigating the origin and formation history of our Galaxy by collecting spectroscopy of representative samples (about 10^5 Milky Way stars) of all Galactic stellar populations, in the field and in clusters. The survey uses globular clusters as intra- and inter-survey calibrators, deriving stellar atmospheric parameters and abundances of a significant number of stars in clusters, along with radial velocity determinations. We used precise radial velocities of a large number of stars in seven globular clusters (NGC 1851, NGC 2808, NGC 4372, NGC 4833, NGC 5927, NGC 6752, and NGC 7078) to validate pipeline results and to preliminarily investigate the cluster internal kinematics. Radial velocity measurements were extracted from FLAMES/GIRAFFE spectra processed by the survey pipeline as part of the second internal data release of data products to ESO. We complemented our sample with ESO archival data obtained with different instrument configurations. Reliable radial velocity measurements for 1513 bona fide cluster star members were obtained in total. We measured systemic rotation, estimated central velocity dispersions, and present velocity dispersion profiles of all the selected clusters, providing the first velocity dispersion curve and the first estimate of the central velocity dispersion for the cluster NGC~5927. Finally, we explore the possible link between cluster kinematics and other physical parameters. The analysis we present here demonstrates that Gaia-ESO survey data are sufficiently accurate to be used in studies of kinematics of stellar systems and stellar populations in the Milky Way.