Jennifer, you mention "all the staples." What you may be looking at is 1/4" underlayment on top of subfloor. If your lucky, all that would need to be cut out is a section of underlayment. If the exposed wood surface is smooth & tight grained, possibly looking like mahogany, and the staples are fairly small (roughly 1/4" wide, point to point) and scattered in a random pattern every 6 to 8", this is underlayment.
The underlay can be cut at any point & does not have to be over a joist. The numerous staples do make it hard to get a starting point, but once you have an edge up it goes better. Remove a section of underlay before cutting through subfloor; it may be just fine. If not, then you'll need to follow up as Tom describes. In replacement, do not use the same joint lines for both subfloor & underlayment; cut the underlay 6" or so away from any joints in the subfloor. Good luck