This work considers the stabilization problem for unstable linear input-delay systems. The main idea of the paper is to use a finite-dimensional approximation for the delay operator, which is based on non-overlapping partitions of the time delay. Subsequently, each individual delay is approximated by means of a classical Pade approximation, where the overall approximation results in a high-order Pade approximation that converges to the original delay operator. By departing from a state-space realization of the approximate process, a linear observer is used to estimate the delay-free output, which is used within a compensation scheme to stabilize the process output. The resulting control strategy has the structure of an observer-based Smith prediction scheme. Numerical results on three examples show that (i) the finer the time delay partition, the better the control performance and (ii) high-order compensators can be required to stabilize certain unstable processes.