Basically, Mrs. Dubose fought her addiction by toughing it out. She just overcame her problem by sheer will power.
One thing that helped her was having Jem and Scout come and read to her. I think that this helped her by giving her something else to think about other than her problems. She spent some time correcting Jem as he read. But much of the time she was just there and did not really seem to be all that "with it."
I think it was really just her own toughness that did it with a little help from the kids because they distracted her.