In this paper we propose a semi-definite programming method for computing outer-approximations (i.e., super-sets) of controlled reach-avoid sets of discrete-time polynomial systems subject to control inputs. The controlled reach-avoid set is a set of all initial states that there exists at least one control policy which steers the system starting from each of them to enter a specified target set in finite time while avoiding a given unsafe set till the target is hit. First, a Bellman type equation, whose unique bounded solution can characterize the exact controlled reach-avoid set, is derived. By relaxing this equation, a set of quantified inequalities for outer-approximating the controlled reach-avoid set is obtained. Via comparing to a set of constraints in state-of-the-art methods on occupation measures, we find that each has its own strengths and can complement each other in outer-approximating controlled reach-avoid sets. As a consequence, we integrate them and obtain a new set of constraints, which is weaker and thus contributes to the gain of tighter outer-approximations. The resulting set of constraints can be encoded into a semi-definite program via the sum-of-squares decomposition for multivariate variables, which can be solved efficiently via interior point methods in polynomial time. Finally, several examples demonstrate the benefits of our method on gaining tighter outer-approximations of controlled reach-avoid sets over existing methods.