If you apply the same force for the same period of time, the linear velocity of the body will be the same in both cases, assuming the body is unconstrained. However, having applied the same force for the same amount of time does not mean that the same amount of energy has been transferred. The energy, or the work done by the force, is the force times the displacement along the direction of the force. This displacement will be greater if the force is not applied through the center of mass.
EDIT: I think the problem is that your intuition tells you that applying a force $F$ to a body for a certain time period $\Delta t$ means that you are transferring energy proportional to $F\cdot\Delta t$. This is not true in the general case. The energy transferred is the work done by the force: $F\cdot d$, where $d$ is the displacement along the direction of the force of the point that the force is applied to. Basically, when you apply the force along the center of mass of the body, the displacement will be smaller, because it corresponds to an equal displacement of the whole body, but when you apply the force further from the center of mass, the displacement will be larger because it is a combination of displacement of the whole body and rotation of the body.