Quantum mechanical investigation of the rotationally inelastic collisions of CS with ortho- and para-H2 molecules is reported. The new global four-dimensional potential energy surface presented in our recent work is used. Close coupling scattering calculations are performed in the rigid rotor approximation for ortho- and para-H2 colliding with CS in the j = 0-15 rotational levels and for collision energies ranging from 10(-2) to 10(3) cm(-1). The cross sections and rate coefficients for selected rotational transitions of CS are compared with the ones previously reported for the collision of CS with He. The largest discrepancies are observed at low collision energy, below 1 cm(-1). Above 10 cm(-1), the approximation using the square root of the relative mass of the colliders to calculate the cross sections between a molecule and H2 from the data available with (4)He is found to be a good qualitative approximation. The rate coefficients calculated with the electron gas model for the He-CS system show more discrepancy with our accurate results. However, scaling up these rates by a factor of 2 gives a qualitative agreement.