Many real-world security problems exhibit the challenge of sequential attacks (i.e., the attacker carries out multiple attacks in a sequential manner) on important targets. Security agencies have to dynamically allocate limited security resources to the targets in response to these attacks, upon receiving real-time observations regarding them. This paper focuses on tackling sequential attacks using Stackelberg security games (SSGs), a well-known class of leader-follower games, which have been applied for solving many real-world security problems. Previous work on SSGs mainly considers a myopic attacker who attacks one or multiple targets simultaneously against each defense strategy. This paper introduces a new sequential-attack game model (built upon the Stackelberg game model), which incorporates real-time observations, the behavior of sequential attacks, and strategic plans of non-myopic players. Based on the new game model, we propose practical game-theoretic algorithms for computing an equilibrium in different game settings. Our new algorithms exploit intrinsic properties of the equilibrium to derive compact representations of both game state history and strategy spaces of players (which are exponential in number in the original representations). Finally, our computational experiments quantify benefits and losses to the attacker and defender in the presence of sequential attacks.