There are basically two answers. One is that, at least according to the Hadith, the call to prayer already existed and would have been called out for the dawn prayer. The second answer is that pre-modern sleep practices were very different than our modern concept of one, approximately eight hour long period of uninterrupted sleep. Pre-moderns, including Muhammad, would more frequently have had a "first sleep", followed by a period of awakeness, followed by a "second sleep." This is also referenced in the hadith: https://sunnah.com/urn/1271150

Separately, in lieu of alarm clocks, there were plenty of animals that make noise at dawn: http://sunnah.com/bukhari/19/13