We prove a central limit theorem for a certain class of functions on sparse rank-one inhomogeneous random graphs endowed with additional i.i.d. edge and vertex weights. Our proof of the central limit theorem uses a perturbative form of Stein's method and relies on a careful analysis of the local structure of the underlying sparse inhomogeneous random graphs (as the number of vertices in the graph tends to infinity), which may be of independent interest, as well as a local approximation property of the function, which is satisfied for a number of combinatorial optimisation problems. These results extend recent work by Cao (2021) for Erdős-Rényi random graphs and additional i.i.d. weights only on the edges.