Cry to sleep

Okay so I'm not letting my baby "cry it out". She is fed, clean, comfortable and lightly sleeping in my arms. I set her down she cries and won't stop until I pick her back up.

I let her cry for a few minutes while I googled what to do and she actually cried herself to sleep.

This is not what I want. I hate that she was so unhappy, but we haven't gotten any sleep in a month. She's awake all night and cries if she's put to bed. This might be our only solution.

I will absolutely respond to every cry when she wakes up, but do you think it's okay to let her fall asleep this way at 2 months? The internet can't make this distinction between crying to sleep and crying it out.