We consider the contact process on a dynamic graph defined as a random $d$-regular graph with a stationary edge-switching dynamics. In this graph dynamics, independently of the contact process state, each pair $\{e_1,e_2\}$ of edges of the graph is replaced by new edges $\{e_1',e_2'\}$ in a crossing fashion: each of $e_1',e_2'$ contains one vertex of $e_1$ and one vertex of $e_2$. As the number of vertices of the graph is taken to infinity, we scale the rate of switching in a way that any fixed edge is involved in a switching with a rate that approaches a limiting value $\mathsf{v}$, so that locally the switching is seen in the same time scale as that of the contact process. We prove that if the infection rate of the contact process is above a threshold value $\barλ$ (depending on $d$ and $\mathsf{v}$), then the infection survives for a time that grows exponentially with the size of the graph. By proving that $\barλ$ is strictly smaller than the lower critical infection rate of the contact process on the infinite $d$-regular tree, we show that there are values of $λ$ for which the infection dies out in logarithmic time in the static graph but survives exponentially long in the dynamic graph.