The contact process on dynamic edges (CPDE) is a contact process evolving on a dynamic environment given by a dynamical percolation on the edges of Z d\,: each edge updates its state to open or closed with respective rates vp and v(1 -p). By coupling a well-chosen subset of once infected sites in the CPDE with a cluster of some supercritical percolation on the edges of Z d , we prove that, for every dimension d $\ge$ 2, we can find some slightly subcritical p < pc(d) such that for every update speed v > 0, the contact process with large enough infection rate can survive. This extends the result for dimension 1 proved by Linker and Remenik in [LR20].