not entirely you wont necesarily eliminate noise by bringing the exposure down, but if you use PP to bring the exposure up it also bring the noise up...thats not a great way to explain it.
Post processing cant differentiate from noise from actual signal so when you adjust your exposure making all the real natural colors brighter, i also increases the brightness of the noise in the pixel making it stand out alot more...im really not sure how to explain it perfectly but its true i swear! and the same in reverse, bringing your exposure down darkens the signal along with the noise making it look cleaner
try it out in photoshop, take your first image and do a 100% crop with the exposure as is, then adjust your exposure and look at the difference in the noise levels
hopefully someone who can explain the reason better than me will chime in here and help me out with the explanation