This is called the Ferber method and it worked very well for us as well.
Note to others considering using it, the method is just as much about the ritual leading up to sleep (bath, reading a book, etc)
“Solve your child’s sleep problems” by Richard Ferber, M.D. is the book which includes this. It also includes a lot more, including an overview of sleep physiology including phases, as well as lots more like details about night terrors and other “partial arousals”.
The big thing he says for going to sleep is to set up the desired sleep associations (eg, alone in the crib in a dark quiet room if that’s what you choose) and then ensure the child falls asleep in that situation and it doesn’t change while they sleep. So for instance, don’t let them fall asleep on your lap then move them to the crib, because when the next normal sleep cycle ends and they partially wake up, they will discover something is different and get upset.
The big thing he says for going to sleep is to set up the desired sleep associations (eg, alone in the crib in a dark quiet room if that’s what you choose) and then ensure the child falls asleep in that situation and it doesn’t change while they sleep. So for instance, don’t let them fall asleep on your lap then move them to the crib, because when the next normal sleep cycle ends and they partially wake up, they will discover something is different and get upset.