We present a memory-based snowdrift game (MBSG) taking place on complex networks. We found that, when a lattice is taken to be the underlying structure, the transition of spatial patterns at some critical values of the payoff parameter is observable for two models of lattices with 4 and 8 neighbors. The transition points can be obtained by local stability analysis, the analytical results are in good accordance with simulations. In sharp contrast to previously reported results, cooperation is promoted by the spatial structure in the MBSG. Interestingly, we figured out that the frequency of cooperation of MBSG on both small-world and scale-free networks is a non-monotonous function of the payoff parameter. This non-trivial phenomenon indicates that giving appropriate advantages to defectors can optimally enhance the cooperation of the system. In addition, we found that the spatial patterns of small-world networks are broken by adding long-range connections. Our work may shed some new light on the evolution of cooperation in networks.