I think it will be hard to draw a straight line between biblical accounts which are most often painted with a brush of possession and things which nowadays are labeled addiction. The closest I've been able to come is with behaviors categorized as self-harm which Psychiatrists are beginning to understand as a form of addiction. If that is the case then the demoniac of the Gadarenes, if examined today, may well be labeled with a self-harm addiction:
Night and day among the tombs and on the mountains he was always crying out and cutting himself with stones. - Mark 5:5
This man is described in the bible as having an unclean spirit and also as being indwelt by many demons. Luke tells us that this man had often been captured, bound, and kept under guard but that he always broke the bonds, escaped the guards, and returned naked to the wilds to wail and cut himself (Mark):
For he had commanded the unclean spirit to come out of the man. (For many a time it had seized him. He was kept under guard and bound with chains and shackles, but he would break the bonds and be driven by the demon into the desert.) - Luke 8:29
If a line can be drawn between this man's behavior and what modern Psychiatry would consider self-harm addiction then the resolution of his affliction may be an example of the biblical basis you seek:
Then people went out to see what had happened, and they came to Jesus and found the man from whom the demons had gone, sitting at the feet of Jesus, clothed and in his right mind, and they were afraid. And those who had seen it told them how the demon-possessed man had been healed. - Luke 8:35-36
Note: I am not suggesting that all mental illnesses or addictions are demon possessions nor am I suggesting that all biblical demon possessions are actually mislabeled mental illnesses.
If it can be accepted (with biblical basis) that Jesus has been granted power and authority to raise the dead, cast out demons, give sight to the blind, restore the lame, heal the sick, etc. and if it can be accepted that the biblical catalogue of miracles that Jesus performed is not a limiting comprehensive list, there should be little difficulty to accept the notion of instant deliverance from heroin addiction (for example) by the power of God.