This would be something of a miraculous "perpetual motion machine," being able to provide a force like that forever. I think it's a great question, though. To clear up the confusion, we need to ask what is meant by escape velocity, which I'll denote v_e.

This term is relevant because we can only provide a push for some finite time. Whether it's a rocket or a cannon ball, you get an impulse and then the projectile is off and moving. If we can't get it moving more than v_e once we stop pushing, then it will move in a parabola and fall back to earth. It might go *really* far, but it will still eventually come back down. If we provide v_e, then it will go up and maintain a stable orbit. It is "falling to earth at the same rate that earth is curving away from it." If, when we stop pushing, we exceed v_e, then it will be moving so fast that, even as earth's gravitational field decelerates it, the rocket has to be infinitely far away before that deceleration alone brings it to v = 0. So it will never come back to earth and has "escaped."

As a sidenote, keep in mind that the force of gravity decreases as you get farther away. So if our magic force generator truly provided a constant force, then "just barely moving" at earth's surface would translate into "rocket speeds" as it gets further away. This thing would just keep moving faster and faster. 1N fighting 0.9N isn't much, but 1N fighting 0.009N is totally different!