The question of whether the central limit theorem (CLT) holds for the total number of edges in exponential random graph models (ERGMs) in the subcritical region of parameters has remained an open problem. In this paper, we establish the CLT. As a result of our proof, we also derive a convergence rate for the CLT, an explicit formula for the asymptotic variance, and the CLT for general subgraph counts. To establish our main result, we develop Stein's method for the normal approximation of general functionals of nonlinear exponential families of random variables, which is of independent interest. In addition to ERGMs, our general theorem can also be applied to other models. A key ingredient needed in our proof for the ERGM is a higher-order concentration inequality, which was known in a subset of the subcritical region called Dobrushin's uniqueness region. We use Stein's method to partially generalize such inequalities to the subcritical region.