Diffusion-controlled A + B --> O reactions in constrained geometries are well-known to obey nonconventional dimension-dependent kinetics. We investigate these kinetics in narrow d-dimensional tubes to study the crossovers from early time d-dimensional kinetics to eventual one-dimensional kinetics. We rely on a reaction-diffusion model that leads to analytically verifiable quantitative results. The crossover times are identified as those times at which the kinetic exponent exhibits maximum curvature and are shown to be universally given by t(c) = 0.034W(2)/D where W is the width of the tube along a narrow direction and D is the diffusion coefficient of both species. Our procedure overcomes many of the statistical difficulties inherent in:Monte Carlo simulations of this problem [Ahn, J.; Kopelman, R.; Argyrakis,. J. Chem. Phys. 1999, 110, 2116].