In this work we prove sufficient conditions for the Glauber dynamics corresponding to a sequence of (non-product) measures on finite product spaces to be rapidly mixing, i.e. that the mixing time with respect to the total variation distance satisfies $t_{mix} = O(N \log N)$, where $N$ is the system size. The proofs do not rely on coupling arguments, but instead use functional inequalities. As a byproduct, we obtain exponential decay of the relative entropy along the Glauber semigroup. These conditions can be checked in various examples, which include the exponential random graph models with sufficiently small parameters (which does not require any monotonicity in the system and thus also applies to negative parameters, as long the associated monotone system is in the high temperature phase), the vertex-weighted exponential random graph models, as well as models with hard constraints such as the random coloring and the hard-core model.