It does not make the tank automatically cycled that particular moment, but the good bacteria on the established filter media will reproduce in the new tank. They will begin to establish on substrate, decorations, etc (any porous surface in the tank). We use this method quite a bit in the fishhouse and with over 130 tanks setup we've yet to have issue with it.
I would leave the filter on the new tank for atleast 1 week, maybe 2 just to be perfectly safe. You don't even have to move the filter, just the filter media, but assuming there is enough surfaces in the already established tank you could setup a new filter on it and the good bacteria on the other surfaces should populate the filter media of the new filter in a few days and the tank shouldn't miss a beat.
Good luck.