We study the crossover between the mean-field and critical behavior of the two-dimensional Bose gas throughout the fluctuation region of the Berezinskii-Kosterlitz-Thouless phase transition point. We argue that this crossover is described by universal (for all weakly interacting |ψ|4 models) relations between thermodynamic parameters of the system, including superfluid and quasicondensate densities. We establish these relations with high-precision Monte Carlo simulations of the classical |ψ|4 model on a lattice, and check their asymptotic forms against analytic expressions derived on the basis of the mean-field theory.