Unfortunately you do need a key to remove the lock cylinder. Just take the new handle to a locksmith and they can either pick the lock and remove the cylinder, or "re-key" the existing lock to work with your key (I just did this recently). On the down side, it may cost you somewhere between 20-50 bucks, but when compared to the cost of an OEM handle it may be worth it.
If you are strapped for cash you can always try to pick the lock yourself via the "impression" method. Basically you get a special "blank" key, insert it and wiggle it hard, remove, and notice the impression left by the discs. File down to the impression contour and repeat. From what I hear this takes a lot of patience and good anger management.