The walking out of situations he had no chance of winning was by far his most powerful ability in the comics. He is extraordinarily powerful, able to challenge gods themselves, but in a kinda unconventional way.

He was from a long bloodline of magic users called The Laughing Magicians and they had the ability to walk "synchronicity waves" and used that to challenge (and at times kill or enslave) gods. As long as Constantine is... going with the flow for want of a better word, doing what the universe wants him to, he's functionally immortal.

Bullets will always miss, enemy plans will go awry. He can literally make his own luck and his power was shown working on, and protecting him from, Satan himself. His twin from another universe mastered the ability and was able to warp reality on a whim, manipulating chance to achieve functionally anything.

John rarely uses this power though and never fully tested it's limits, instead relying on more mundane abilities and occasionally magic.