How am I able to run a small dc motor with just a 9v battery?
Batteries actually tend to give much more current, unless it is a very small battery. 
For your motor you need a battery that can supply 9v or more, to know the time it will work you need to know the current drawn from the motor in load conditions and the mAh rating of your battery, lets say your motor draws 2A and you battery has a capacity of 4000mAh, you divide the capacity with the current 4Ah/2A which gives 2 hours of continuous function.