We measure the light-driven response of a magnetic multilayer structure made of thin alternating layers of cobalt and platinum at the few-femtosecond timescale. Using attosecond magnetic circular dichroism, we observe how light rearranges the magnetic moment during and after excitation. The results reveal a sub-5 fs spike of magnetization in the platinum layer, which follows the shape of the driving pulse. With the help of time-dependent density functional theory, we interpret the observations as light-driven spin injection across the metallic layers of the structure. The light-triggered spin current is strikingly short, largely outpacing decoherence and dephasing. The findings suggest that the ability of shaping light fields in refined ways could be translated into shaping new forms of spin currents in materials.