Without any information on what the mousepad is made of, it's impossible to know, but I find it unlikely that water actually damages the material itself. Rather, damage and loss of surface properties from cleaning with water is probably due to two things: (1) mechanical shear resulting from scrubbing damages whatever fabric weave is responsible for the surface properties (Kind of like how they tell you to blot, not scrub, a stain out of a carpet or furniture upholstery); (2) residue from hard water or detergents remains on the surface, which also changes surface properties. Your sweat is mostly water, so the mousepad is getting wet anyway, and so is just about any cleaning agent you are likely to find. Pure solvents like alcohol are more likely to do lasting harm than water is, particularly if the material has some kind of hydrophobic surface coating.
My advice would be to use a wet, but not drenched, soft cloth and blot the pad clean. Then blot dry. I'd use distilled water instead of tap water, and I would try to avoid detergents where possible.
Not 100% sure what product you have, but I found the following link from Razer about their mouse pads and cleaning:https://support.razer.com/gaming-mice-and-mats/razer-goliathus-control-edition/
They allow for cleaning with water, just not fully submerging it.
If what you have is the Golianthus brand, I looked and couldn't find what the fabric actually is, but the promotion website indicate only that it is a microtextured fabric. The microtexture (small strand diameter, tight weave) is what gives the surface low friction properties.https://www.razer.com/gaming-mouse-mats/razer-goliathus-speed
Most microtextured fabrics are made from artificial materials like Nylons or polyesters. These would not be damaged unless they were allowed to soak for a long time. Nylons and polyesters can and do swell in the presence of water, which would damage the weave, but you'd have to expose them to a lot of water and over a long period of time and/or at high temperature. I have microfiber glass cleaning cloths that I use all the time to clean optics at work, and they get damp, and they're fine.https://en.wikipedia.org/wiki/Microfiber
So: were it my pad, I'd use as little water as possible and don't scrub. Alternatively, you'll have to use a dirty mat, which probably will disrupt the low friction properties as much as any slight weave damage will. You may also want to spot test the cleaning process on the corner or something first.