This article investigates distributed finite-time optimization problems for first-order multi-agent systems (MASs) with time-varying cost functions under arbitrary strongly connected digraphs. The objective is to seek the minimizer of the sum of cost functions within finite-time, where each cost function is only known by its corresponding agent. In order to realize the above optimization goal, by using finite-time stability theory and graphs theory, we propose a new class of distributed finite-time optimization algorithms which are independent of Hessian information and the partial derivatives of gradient with respect to time. Furthermore, as an application, the proposed distributed algorithms are extend to handle distributed finite-time economic dispatch problems of smart grids, where both the cost functions and loads are time-varying. Finally, simulation examples are illustrated to verify the theoretical results.