One way to accomplish this is to apply a herbicide labeled for this use to the freshly cut rose canes. Herbicides with triclopyr as the active ingredient are often used for this, and only a tiny amount is needed. This will kill the root of the plant.

Or you can wait. In parts of the midwest and spreading a mite was released to carry a fungus that kills multiflora roses. It has already killed mine. The kicker is that it also kills garden roses that have multiflora roses in their heritage.