We generate an amorphous MgCu model using the rapid solidification of the melt through a first-principles molecular dynamics approach within a generalised gradient approximation and reveal, for the first time, its structural features and mechanical properties in details. The liquid and glassy MgCu are found to acquire slightly distinct local structures. Yet in both forms of MgCu, most Cu atoms have a tendency to form the ideal and defective icosahedrons while Mg atoms are arranged in complex configurations. The mean coordination number of Cu and Mg at 300 K is 11.31 and 13.73, respectively. The short-range order of MgCu glass is projected to be different than the known crystalline MgCu and Mg2Cu phases. The mechanical properties of MgCu glass and the CsCl-type MgCu crystal are computed and compared. On the basis of the enthalpy analyses, a possible pressure-induced crystallisation of the MgCu glass into a CsCl-type structure is proposed to occur at around 11 GPa.