Maybe in your Texas phonetics, Myridon. El-Oh-El.You can be lulled from an excited state to a calmer state. It doesn't necessarily imply sleepiness.
(Said aloud in an Internet context, "lull me" might be confused with "LOL me." )
I agree on the whole ("to lull someone into complacency [before attacking them]"), but I can also see it as meaning "His conversation gradually made me drowsy".I think lull into a false sense security is the commonest way I hear lull used.
If I want to add the notion of sleep to it, I have to add the word sleep to it:
His conversation gradually lulled me to sleep.
His conversation gradually lulled me. would mean something like "His conversation gradually made me feel calm(er)."