Variables affecting the passivation capability of Cu(Mg) alloy films, which were sputter deposited from a Cu (4.5 at. %) target, have been investigated. The results show that the passivation capability of a Cu(Mg) alloy film is a function of annealing temperature, $O_2$ pressure, and Mg content in the film. Increasing the annealing temperature up to $500^{circ}C$ favors formation of a dense MgO layer on the surface which has a growth limited thickness of 150 $AA$. Decreasing the $O_2$ pressure enhances the preferential oxidation of Mg over Cu. Furthermore, increasing the Mg content in the Cu(Mg) film promotes formation of a dense MgO layer. Vacuum pre-annealing was found to be very effective in segregating Mg to the surface, facilitating the passivation capability of the Cu(Mg) alloy film even when the Mg content is low. In the current study, self-aligned MgO layers with low resistivity and an effective passivation capability over the Cu surface, have been obtained by manipulating these factors when Cu(Mg) thin films are annealed.