Raising ISO does not increase noise. Capturing fewer photons does. ISO does not affect how many photons are captured. That's down to how much light there is and your chosen aperture and shutter speed.
In fact, for any given number of photons captured raising the ISO will reduce noise. In other words it is better to shoot a correct exposure at 1600 ISO than to shoot at 800 ISO and underexpose by 1 stop or to shoot at 400 ISO and underexpose by 2 stops and so on.
Here's an example from a sequence I shot a few minutes ago. I metered off the grass and it read 0 on the meter at 400 ISO. Without adjusting shutter speed or aperture (the controls which affect how much light is recorded) I took several shots simply varying ISO from each one to the next. So at 100 ISO the grass was 2 stops under. At 200 ISO at was at 1 stop under, and so on up till 3 stops over at 3200 ISO.
Here is a comparison of the 100 ISO shot with the 3200 ISO shot. I've used auto tone in Lightroom to equalise (more or less) the appearance of the shot. To my eyes the version shot at 3200 ISO is actually cleaner, and definitely with less banding, than the shot that was underexposed at 100 ISO. This does not come as a surprise to me.
(Click the size bar for a proper look.)
It's not high ISO that is the problem. It's underexposure. Raising the ISO actually helps a little.
Put another way, select the shutter speed and aperture you want and then choose the highest ISO you can that will not result in clipping of important highlights. That said, there are some ISO values that should be avoided on some cameras so pick your ISO wisely. However, do not think that keeping it artificially low will reduce noise. If doing so results in an underexposed image then you'll be worse off.
High ISO gets a bad name for noise because people combine it with fast shutter speeds and low light - they don't capture enough photons. It's not the ISO choice that is the problem. It's failure to saturate the sensor. In those situations high ISO is your friend. Don't be afraid to use it.
By the way, all this is related to the good practice of ETTR (expose to the right). If you're already against the wall on shutter speed, aperture and available (or supplementary) light then the only thing left to increase in order to ETTR is your ISO. Do it.