A set of classic molecular dynamics simulations at a cooling rate of 0.1 K/ps have been performed to investigate the effect of pressure ranging from 0 to 4 GPa on the solidification of liquid Mg70Zn30 alloy, by means of the average atomic energy, the largest standard cluster analysis and 3D visualization. It is found that pressure plays an important role in both the glass transition and the structure of the final solid. T-g-P (T-g is the end temperature of the glass transition) is a monotonically increasing curve with the increase rate decreases significantly at P > 0.1 GPa. However, the structure parameters based on short-range order and icosahedrons are not monotonically dependent on pressure. Interestingly, the pressure dependence of the structure parameters based on topologically close-packed (TCP) structures is highly consistent with T-g-P. Therefore, TCP is an essential characteristic and plays an important role in glass transition. In addition, the pressure enhances the contribution of Mg atoms to the formation of Zn-rich TCP structures. These findings shed new light on understanding the pressure-structure relationship of metallic glasses.