I'm no expert at lenses, but I believe that higher-priced lenses (ie. the ones with f/2.8) are better built than cheaper lenses. So even at smaller apertures (bigger f-numbers), you should still get better image quality with the higher-priced lenses.
For example, according to the-digital-picture.com, the kit lens suffers from lower contrast, chromatic aberrations in the corners, and mild vignetting. There are also times where the cheaper lenses aren't as sharp as the more expensive lenses, too. Here's the review:
http://www.the-digital-picture.com/Reviews/Canon-EF-S-18-55mm-f-3.5-5.6-IS-Lens-Review.aspx
(the above link also provides a mouse-over comparison of photos taken from different lenses at the same aperture, so you can compare the quality of photos yourself)
One other thing about lower-priced lenses. Usually, they have variable maximum apertures (ex. f/3.5-5.6). So when you're at 55mm on the kit lens, for example, your maximum aperture is f/5.6. The autofocus (AF) systems in most DSLRs require a minimum aperture of f/5.6 to work. The larger your maximum aperture, the better your AF system will perform. So, I would expect the f/2.8 lens to have a better AF performance at the same 55mm focal length. (Note: this is regardless of what aperture you're using for exposure. It's the
maximum aperture of the lens that affects AF performance)
Plus, the viewfinder will be 4 times brighter (2 stops brighter) with the f/2.8 lens than the cheaper lens at f/5.6. (the viewfinder brightness depends on the maximum aperture) This becomes important if you need to manually focus or if you're using Live View.
One final thought. Typically, the "sweet spot" for any lens is about 1-2 stops below its maximum aperture. The "sweet spot" is the aperture at which your lens produces the best, sharpest images. This varies depending on the lens, but I believe it's a general rule-of-thumb.
Also, because of a concept called
diffraction (for more info, see here and here), your sharpest pictures will be at around f/8 to f/11. Any smaller than that (ex. f/16, f/22, etc), and your pictures will show some softness due to diffraction. Diffraction depends on the sensor of your camera, not on the lens.
Well, this was a much longer reply than I had anticipated. Like I said, I'm no expert at lenses, so I'm sure I've made some mistakes above, so please correct me if I'm wrong.
BTW, I guess my answer to your question is: yes, higher-priced lenses (f/2.8 lenses) will give you better quality images than kit lenses, even at smaller apertures.