Polymerization of rigid rodlike molecules with reactive end groups (e.g., poly(p-phenylene terephthalamide)) requires near parallel orientation of the molecules. The reaction becomes diffusion-limited as the rotational diffusivity of the reacting molecules decreases to low values in the later stages of the reaction, and this ultimately limits the molecular weight of the polymers formed. Here a theoretical study of the step-growth reaction between rodlike molecules in a solution under extensional flow is carried out. A model for the process is developed using the Smoluchowski approach which yields the effective reaction rate constant for the process in terms of the system parameters. The rate of reaction increases with extensional rate because of flow-induced orientation of molecules, and for sufficiently high extensional rates, the rate constant becomes higher than the intrinsic value. This is an instance where flow has a catalyst-like effect on the reaction. The results indicate that flow can be a useful tool for accelerating reactions and thus improving yield and selectivity for orientation-dependent reactions between large anisotropic molecules.