—When did it begin to snow?
—It started by the night.
Is the underlined part a natural collocation? I think "It started during the night" would be better.
Am I right?
"It started by the night", although it's not a good sentence, could mean "It was snowing before night fell."
"It started during the night" is grammatical; but it's only better if that's your meaning.
I think 'by' here would require the past perfect,
It had started by night(fall).
"It started overnight" is an option as well. Depending on your exact meaning.