Salamu alaykum,
The occurrence of night and night is not tied with the appearance of the moon.
Does the Quran say's otherwise.
The night and day happen due to earths rotation along its axis. The side facing the sun is the one doing through daytime and the side not facing the sun is the one experiencing night. So only one side of the earth can be facing the sun which corroborates that its either day or night, cannot be both at the same time
By the sun and his brightness, And the day when it revealeth him (The Sun).
By the sun and his brightness, By the Night as it conceals it (The Sun).
Does the verse say that moon unveils the night? It's the day that unveils the Sun and the Night that conceals the Sun.
Now compare your statment with the verses of the Quran.
wasalamu alaykum