With my AAdvantage Credit Card will the process of me getting the miles be when the 1st statement due date arrives or when I pay off the credit card?

I met my cards minimum spend in the first month but it show zero dollar owed and zero minimum payment needed but I want to get my miles asap (to book a flight) so just wondering in order to get the miles to show up in my account do I need to pay off the credit card balance even though it says I do not need to.

thanks,

John