Using the "dispensed drop" variant of the sessile drop technique, spreading kinetics of dilute Cu-Cr alloys on smooth vitreous carbon substrates are measured under helium microleak conditions. In this system, it is known that the drop spreading rate is controlled by diffusion of the reactive atom species (Cr) from the bulk liquid to the triple line, where wetting is induced by formation of an interfacial layer of chromium carbide. Microstructural characterization of rapidly cooled drops shows that growth of the interfacial reaction product layer continues behind the moving solid-liquid-vapor triple line. The spreading velocity is modeled by finite-difference numerical analysis of diffusion near the triple line in the presence of continued interfacial reaction, simplifying the growth rate as being constant and using realistic parameter values. We show that continued interfacial reaction explains the dependence of the triple line spreading rate on the instantaneous wetting angle that is observed in this system.