We consider variational problems with path-dependent terminal costs. Motivated from Mogulskii’s theorem in large deviation theory and dynamic importance sampling for path-dependent rare events, we focus on particular forms of Lagrangians with superlinear growth. By reformulating the variational problem to a value function of a path-dependent deterministic control, we study it by path-dependent dynamic programming methods. Under co-invariant derivative notion on path spaces, the value function is related to a Hamilton-Jacobi partial differential equation (PDE) with a path-dependent terminal condition. Using a viscosity type solution proposed by Lukoyanov for a weak notion, we show that the value function can be characterized as a unique viscosity solution of the Hamilton-Jacobi PDE.