It depends on your definition of exposure. Exposure is the overall amount of light that falls on the film or sensor. So by that definition, if you have a fixed shutter speed, then if you use a wider aperture, you'll let in more light, so you will increase the exposure. A narrower aperture opening will let in less light, so less exposure.
Keep in mind that a camera on Auto mode will adjust, so if you narrow the aperture, the camera will slow the shutter to compensate, and you'll end up with roughly the same exposure. So for aperture to affect exposure, you have to keep other parameters constant.
People also use the term exposure as a substitute for shutter speed, as in "I took a long exposure of the stars". If you are thinking exposure = shutter speed, then I can understand your confusion. But the real definition of exposure is the overall amount of light. This is affected by the exposure triangle of ISO, aperture and shutter speed.