I discovered this for myself by using Benzoyl Peroxide 10% as an antiseptic on a mosquito bite which looked as if it would get infected.
A blob of cream onto the bite, then this was covered by a sticking plaster overnight - because Benzoyl Peroxide is a bleach, and will bleach your clothing, bedding or towels if it is not covered, then washed-off carefully.
Within less than an hour, the bite had stopped itching, and the next day it had started to flatten-out - I generally had no further trouble.
To provide context, a mosquito bite will usually last me for 4-6 weeks, and itch for most of that time.
This first accidental experiment seemed promising; so I tried it a few more times over the summer (which was exceptionally warm, humid and mosquito-y), and it works for me as described above. Once I needed an extra second overnight treatment when a bite began itching again.
I've also found a couple of similar experiences reported on the interweb - so it seems pretty conclusive. Mosquito bites can apparently be cured with 10% Benzoyl Peroxide - which product is obtainable without prescription - it is mostly used to treat acne.
Note: I previously reported that BP can also be used to treat shaving rash/ ingrowing beard stubble...
Note added: In the comments; CCL describes hypothetical possible mechanisms by which BP might work to help mosquito bites. Conventionally BP is supposed to have a dual-action - an oxidising antiseptic in the short-term and and a peeling agent over a few days. But the rapidity of the action I describe, suggests to me that BP is rapidly denaturing the mosquito saliva to prevent its irritating effect, and also disabling the local inflammatory immune response... in some way.