—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 depends on when the snow started. 'By' usually means 'before', as in "Please do it by Tuesday."
"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.