It is shown that the exothermic first-order reaction A --> B + heat may spatiotemporally self-organize through the loss of stability of its stationary state owing to recently discovered differential-flow-induced chemical instability. The heat released in the reaction acts autocatalytically (as an activator), and the reactive fluid represents the inhibitor. A porous packing in a tubular cross-flow reactor retards the heat flow relative to the flow matter, introducing the necessary differential flow of the key species. This disengages the activator from the local inhibitor response and allows the tendency of growth inherent in the activator subsystem to destabilize the system as a whole. The dynamic consequence of the instability is the formation of temperature and concentration waves moving through the reactor. This mechanism of pattern formation applies to the entire class of exothermic reactions.