The Gromov-Wasserstein (GW) distance quantifies discrepancy between metric measure spaces, but suffers from computational hardness. The entropic Gromov-Wasserstein (EGW) distance serves as a computationally efficient proxy for the GW distance. Recently, it was shown that the quadratic GW and EGW distances admit variational forms that tie them to the well-understood optimal transport (OT) and entropic OT (EOT) problems. By leveraging this connection, we establish convexity and smoothness properties of the objective in this variational problem. This results in the first efficient algorithms for solving the EGW problem that are subject to formal guarantees in both the convex and non-convex regimes.