We present a kinematic analysis of the globular cluster (GC) system in the giant elliptical galaxy (gE) M60 in the Virgo Cluster, using a photometric and spectroscopic database of 121 GCs (83 blue and 38 red GCs). We have found that the M60 GC system shows a significant overall rotation. The rotation amplitude of the blue GCs is slightly smaller than or similar to that of the red GCs, and the position angles of their rotation axes are similar. The velocity dispersions about the mean velocity and about the best-fit rotation curve for the red GCs are marginally larger than those for the blue GCs. Comparison of observed stellar and GC velocity dispersion profiles with those calculated from the stellar mass profile shows that the mass-to-light ratio should increase as the galactocentric distance increases, indicating the existence of an extended dark matter halo. The sample of all the GCs in M60 is found to have a tangentially biased velocity ellipsoid, unlike the GC systems in other gEs. The two subsamples appear to have different velocity ellipsoids. The blue GC system has a modestly tangentially biased velocity ellipsoid, while the red GC system has a modestly radially biased or an isotropic velocity ellipsoid. We compare the kinematic properties of the M60 GC system to those of other gEs ( M87, M49, NGC 1399, NGC 5128, and NGC 4636), and discuss the implication of these results for the formation models of the GC system in gEs.