Many industrial processes can be approximated by low order plus dead-time models. In this work we propose a novel Generalized Predictive PI (GPPI) controller to achieve fast over-damped responses for systems with long dominant dead-times. The already established Predictive PI (PPI) control strategy has demonstrated to exceed the traditional PID controllers when they are applied to systems with long dominant dead-time, but the PPI structure limits the closed loop design options, thus also limiting the achievable performance. Here we analyze the design of the proposed GPPI controller versus the PPI and PI controllers to achieve fast over-damped responses. An expression for the achievable performance in terms of the Integral Absolute Error (IAE) is obtained for the GPPI controller and used to compare with the other controllers. Tuning rules are also proposed and simulation examples provided to offer some additional insight into the use of the GPPI controller.