Near optimal evasion strategies from pursuers employing modern linear guidance laws, which require the evader’s acceleration, are proposed. To attain small miss distances, most modern guidance laws use the evader’s acceleration. However, the evader’s acceleration can not be directly measured and needs to be estimated. The key idea underlying the proposed approach is to exploit the inherent time-delay associated with the estimate of the evader’s acceleration to maximize the miss distance. The problem is posed in a linear bounded acceleration optimal control framework with arbitrary order linear pursuer and evader dynamics. The evader is assumed to have perfect information on the pursuer’s states, parameters, and guidance law. The pursuer is assumed to have perfect information on the evader’s dynamic parameters and states, however, the pursuer’s estimate of the evader’s acceleration is assumed to have a pure delay. The optimal evasion strategies are derived and can be computed in real-time by a simple integration in reverse time. Simulation results demonstrate the viability of the proposed evasion strategies. It is shown that the miss distances achieved using the proposed strategies, against pursuers employing linear guidance laws that use target acceleration, are substantially higher than the miss distances archived without exploiting the inherent estimation delay.