We investigate the ground-state properties of a dual-species spin-1/2 Bose-Einstein condensate. One of the species is subjected to a pair of Raman laser beams that induces spin-orbit (SO) coupling, whereas the other species is not coupled to the Raman laser. In certain limits, analytical results can be obtained. It is clearly shown that, through the interspecies spin-exchange interaction, the second species also exhibits SO coupling. This mixture system displays a very rich phase diagram, with many of the phases not present in an SO-coupled single-species condensate. Our work provides a way of creating SO coupling in atomic quantum gases, and opens up an avenue of research in SO-coupled superfluid mixtures. From a practical point of view, the spin-exchange-induced SO coupling may overcome the heating issue for certain atomic species when subjected to Raman beams.