Not only can it be done, it has been done. But how, might you ask. Well, it hasn't been done using a rope, but it has been done using gravity, and it's done all the time. That's the basic premise of a Gravity Slingshot, come close enough to a big object to have it help you change your course.
In the end, this will only change your velocity with respect to the outside world. Your speed relative to the Asteroid will be more or less the same after. But you will change speed with respect to the outside world.
As for the use of the tether itself, it would have to be very strong, and have a very strong grip. You'd have to get it placed very close to the target, and let go at the precise moment, or else it would spell disaster. But it could be done, in theory, and would save some fuel, if you can make everything line up perfectly. But any object small enough to allow for this to happen would be difficult to get with the required precision to intercept it, most likely losing most of the gain you would get.