In chapter 11, Jem destroys Mrs. Dubose's camellia bush in a fit of rage, and Atticus forces his son to read to Mrs. Dubose for two hours each day, except on Sundays, for an entire month as punishment. Mrs. Dubose pases away a short time after Jem's punishment is over, and Atticus explains to his son that she suffered from a chronic illness. He proceeds to tell Jem that Mrs. Dubose's chronic illness was extremely painful and she became addicted to morphine to alleviate her pain. Atticus says that before Mrs. Dubose passed away, she wanted to conquer her morphine addiction and "leave this world beholden to nothing and nobody" (Lee, 115). He agreed to help Mrs. Dubose conquer her morphine addiction by making Jem read to her each day between her doses of morphine. Jem's reading kept Mrs. Dubose's mind occupied and off her pain long enough between her morphine doses until she was able to break her habit. Mrs. Dubose passes away shortly after conquering her morphine addiction and Atticus tells his children that she was the bravest person he's ever known.
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.