We investigate the controllability from the exterior of space-time fractional wave equations involving the Caputo time fractional derivative with the fractional Laplace operator subject to nonhomogeneous Dirichlet or Robin type exterior conditions. We prove that if 1 < alpha < 2, 0 < s < 1 and Omega subset of R-N is a bounded Lipschitz domain, then the system D-t(alpha) u + (-Delta)(s) u = 0 in Omega x (0,T), Bu = g in (R-N\Omega) x (0,T), u(center dot,0) = u(0), partial derivative(t)u(center dot,0) = u(1) in Omega, is approximately controllable for any T > 0, (u(0), u(1)) is an element of L-2(Omega) x V-B(-1/alpha) and every g is an element of D(O x (0,T)) where O subset of (R-N\Omega) is any non-empty open set in the case of the Dirichlet exterior condition B-u = u, and O subset of R-N\Omega is any open set dense in R-N \ Omega for the Robin exterior conditions Bu := N(s)u + kappa u. Here, N(s)u is the nonlocal normal derivative of u and V-B(-1/alpha) denotes the dual of the domain of the fractional power of order 1/alpha of the realization in L-2(Omega) of the operator (-Delta)(s) with the zero (Dirichlet or Robin) exterior conditions Bu = 0 in R-N \ Omega.