Bedfellows.

A good way to do what you're attempting is to start with a NURBS plane and a basic NURBS circle. Project the curve onto the surface of the plane and trim the center of the circle. Convert the NURBS geometry to Polygon and adjust the settings in the bottom of the channel box(sorry to be so vague here, I'm not looking at Maya right now.) When you initially convert it, it will look like garbage. But the history connection, adjusted with the settings at the bottom of the channel box, will allow you to optimize the conversion after the fact. By increasing the sections and spans in the NURBS history info, you will also be increasing your edge count in the poly shape until the shape is what you originally intended. Once you have the Poly shape you can extrude it like normal.

This is one of the things I really don't like about 3D. NURBS patches are a pain in the a$$ to attach, most of the time, but NURBS are the only geometry type that will give really nice flowing curves/creases AND completely flat surfaces around them. Booleans work, mut most of the time they make the geometry "Heavy" when you use them too much and after one boolean operation on one particular object, typically, you can't perform any more without a work around. FormZ actually is a little better with these types of operations, Amapi seems like it might be better as well, but these processes benefit greatly from using NURBS and having more robust NURBS tools.

Another way, in Maya, could be to use the create poly tool to first build the plane and then, it's either cntrl or option, to create a subtractive shape. Basically it will allow you to build the geometry with a hole in it, but I don't think there's a way to build a shape that isn't open with that method.