How to deal with this heavily underexposed footage
If this is no raw material, there is no additional pixel information to balance your colors and achieve a natural look. Only chance you have is to denoise your video and desaturate the colors. Try to find a good relationship between brightness and noise to get an acceptable result.

Unfortunately there is no good, free, build-in or open source solution at the moment. Denoising a video requires very complex algorithms, not only because every camera produces different noise types. Currently best approach are wavelet algorithms, but there are only 2 tools which use it:


Neat Video
Nuke's Denoiser


Note: In my experience Neat Video does a very good job and in some cases you are able to denoise the video twice without getting an over blurred image.