I have two kids, and I've had plenty of sleepless nights that have left my brain feeling dull. Both of my kids woke up in the middle of the night for what felt like eons - but I have never had a problem getting them to fall asleep.
Weird thing about humans: They don't know how to fall asleep on their own. Sleep is a learned skill.
Before I had a baby, I did loads of research on infant sleep. I attended multiple lectures from sleep experts and read lots of books and websites, and they all told me the same thing:
Put the baby down while they're drowsy, but awake.
I followed that advice from the beginning, and it worked! Every time my kids started to look sleepy (eye rubbing and general crankiness are tell-tale signs), I just put them down in their crib, and generally, they would fall asleep a few minutes later.
It was that simple. Once they learned to sleep on their own, I never had to fight with them at bedtime.
The biggest sleep mistake new parents can make, in my opinion, is rocking their kids to sleep.
Of course there are times that it feels so nice to have a snuggly baby fall asleep on your chest - and I've done that plenty of times - but ultimately if you want your baby to learn to sleep on their own, you should put them down.