We present an analytic theory of quantum criticality in the quasi-one-dimensional topological Anderson insulators of class AIII and BDI. We describe the systems in terms of two parameters (g, χ) representing localization and topological properties, respectively. Surfaces of half-integer valued χ define phase boundaries between distinct topological sectors. Upon increasing system size, the two parameters exhibit flow similar to the celebrated two-parameter flow describing the class A quantum Hall insulator. However, unlike the quantum Hall system, an exact analytical description of the entire phase diagram can be given. We check the quantitative validity of our theory by comparison to numerical transfer matrix computations.