[masaegu]

Yes, shinjitsu, they are the same word. The verb 'yasumu' means to take a break, to rest, to be absent, etc.. You have incidentally asked a good question, too. For some reason, we seldom write 'Oyasuminasai' with a kanji even though it's a very simple kanji. Same thing with 'Ohayou' or 'Konnichiwa'. These phrases only require pretty simple kanji but we usually write them all in Hiragana. If I may take a wild guess, I think this is because Japanese parents (and teachers) want their kids to be able to write these important everyday phrases as soon as possible, that is before they learn the necessary kanji in school.