We present a kinematic analysis of the globular cluster (GC) system in the giant elliptical galaxy (gE) M60 in the Virgo cluster. Using the photometric and spectroscopic database of 121 GCs (83 blue GCs and 38 red GCs), we have investigated the kinematics of the GC system. 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 their angles of 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 be increased as the galactocentric distance increases, indicating the existence of an extended dark matter halo. The entire sample of GCs in M60 is found to have a tangentially biased velocity ellipsoid unlike the GC systems in other gEs. Two subsamples appear to have different velocity ellipsoids. The blue GC system has a modest tangentially biased velocity ellipsoid, while the red GC system has a modest radially biased or an isotropic velocity ellipsoid. From the comparison of the kinematic properties of the M60 GC system to those of other gEs (M87, M49, NGC 1399, NGC 5128, and NGC 4636), it is found that the velocity dispersion of the blue GC system is similar to or larger than that of the red GC system except for M60, and the rotation of the GC system is not negligible. The entire sample of each GC system shows an isotropic velocity ellipsoid except for M60, while the subsamples show diverse velocity ellipsoids. We discuss the implication of these results for the formation models of the GC system in gEs.