  We report a combined experimental and theoretical study of the unusual
ferromagnetism in the one-dimensional copper-iridium oxide Sr$_3$CuIrO$_6$.
Utilizing Ir $L_3$ edge resonant inelastic x-ray scattering, we reveal a large
gap magnetic excitation spectrum. We find that it is caused by an unusual
exchange anisotropy generating mechanism, namely, strong ferromagnetic
anisotropy arising from antiferromagnetic superexchange, driven by the
alternating strong and weak spin-orbit coupling on the $5d$ Ir and 3d Cu
magnetic ions, respectively. From symmetry consideration, this novel mechanism
is generally present in systems with edge-sharing Cu$^{2+}$O$_4$ plaquettes and
Ir$^{4+}$O$_6$ octahedra. Our results point to unusual magnetic behavior to be
expected in mixed 3d-5d transition-metal compounds via exchange pathways that
are absent in pure 3d or 5d compounds.
