"...Over a period of time, the asteroid is deflected from its original course..."
"...Only if the thing is not rotating..."
Actually, yes, even if the thing is rotating, although that makes the energy requirements of the laser go up somewhat, but not much.
The laser heats a spot where you want the "jet" to occur, and with enough laser power, that happens in just a few seconds. As the asteroid rotates beneath the laser beam, other areas of the rock are vaporized, but the direction of the impulse remains the same. It would just create a linear scar on the asteroid, instead of a simple "hole" of ejected material. The key is that the craft delivering the laser is at "station keeping".
You can even use the laser to first stop the asteroid's tumbling, if desired.
Here is a link to a video testing a laser on a small rotating rock, to illustrate the principle:
Small Scale Test of Laser Ablation