The Szeged index of a graph is an invariant with several applications in chemistry. The power graph of a finite group $G$ is a graph having vertex set as $G$ in which two vertices $u$ and $v$ are adjacent if $v=u^m$ or $u=v^n$ for some $m,n\in \mathbb{N}$. In this paper, we first obtain a formula for the Szeged index of the generalized join of graphs. As an application, we obtain the Szeged index of the power graph of the finite cyclic group $\mathbb Z_n$ for any $n>2$. We further obtain a relation between the Szeged index of the power graph of $\mathbb Z_n$ and the Szeged index of the power graph of the dihedral group $\mathrm{D}_n$. We also provide SAGE codes for evaluating the Szeged index of the power graph of $\mathbb{Z}_n$ and $\mathrm{D}_n$ at the end of this paper.