Assuming all other exposure settings with and without flash are equal, then using flash means you are adding light to the scene. Increased light in the scene means increased light down the lens, which means more light at the sensor. That means you have a higher signal to noise ratio at the sensor, which generally means less noise.

Signal to Noise ratio, or S/N, is extremely important to the amount and quality of noise in digital photography. Every pixel in a sensor has a certain range of luminance it supports, from the read noise floor (not quite pure black, but close to it) to pure red, green, or blue (at ISO 100 this is called full well capacity, and generally termed maximum saturation at all ISO settings). A pure red, for example, would be maximum S/N for that pixel. A pixel of an 18% gray card would be roughly 50% S/N. The lower the signal, the more apparent noise (of which photon noise dominates) will be. The higher the signal, the less apparent noise will be.

Any time you can add light to your scene, be it flash, other artificial light, or even sources of natural light, will usually improve the signal to noise ratio. At higher ISO, your maximum saturation will be a fraction of the full well capacity, which is almost the same as underexposing (at ISO 100) one stop for each stop of ISO, then pushing exposure in post by the same amount. If you underexpose at ISO 100 by two stops then boost by two stops in post, that would be similar to using ISO 400 (with the added caveat that you'll boost read noise as well when pushing in post, but let's assume an "ideal" sensor for the sake of simplifying discussion.)

If you underexposed by three stops, then pushed in post, that would be similar to using ISO 800. Before pushing in post, you'll notice that the image is pretty dark. If you do the same thing, underexpose by three stops at ISO 100...but add flash...the unmodified image in post should be a bit brighter than the first shot that was underexposed by three stops. Flash adds light, which increases signal level. Boost both images such that they appear the same post, and the second shot should appear less noisy, even if only by a small margin....it will require LESS of a boost to exposure to correct.