This paper studies global synchronization between a heterogeneous dynamical network and a known target trajectory via distributed impulsive control. Synchronization with an error level, called quasi-synchronization, is analyzed by utilizing the time-varying Lyapunov function. Some sufficient quasi-synchronization conditions are presented and explicit expressions of error levels are derived. Furthermore, the effects of the pinning control matrix and the coupling strength are explored. Unlike the continuous pinning feedback control, it is shown that a large coupling strength will destroy synchronization in the present of impulsive controller, which is also verified by our simulations.