In this paper, a new type of fractional-order multi-agent systems (FOMASs) with parameter uncertainties and Takagi-Sugeno (T-S) membership functions is developed on a directed communication topology graph. Unlike existing results that either adopt fixed-gain impulsive control or deal with fractional-order consensus without simultaneously considering communication efficiency and parameter uncertainties, we propose a codesigned T-S fuzzy (TSF) impulsive control framework. By collecting instantaneous local information, leader-following and leaderless distributed TSF impulsive controllers are designed to realize the exponential consensus for the proposed FOMASs, where the impulsive gain is dynamically adjusted by the same set of fuzzy rules describing the system. In addition, some sufficient conditions with regard to the exponential consensus of FOMASs are addressed in terms of linear matrix inequalities (LMIs). Compared with the traditional impulsive controllers with the fixed gain, the designed TSF impulsive controller has faster convergence speed to achieve the exponential consensus of FOMASs with uncertainty, and it has lower energy consumption. We then present an application example in a single-link manipulator to validate the practical efficacy for the proposed theoretical findings.