How does Lawrence manage to make the events in the story believable, even though there are clearly supernatural elements at play?
There certainly is the indication of supernatural occurrence in this story - all under the guise of "luck." The reason the story is believable and something more than a story about luck and Paul's prophetic picks is that the story is more of an allegory about the inability of love to be supplanted by money.
Supernatural or not, Paul's ability to predict winners is either divinely inspired, a deal with the devil, or just plain dumb luck. Given the story's allegory which is a criticism of materialism (money), it would seem that either luck or a sinister force of the supernatural is at work. Paul is simply trying to please his mother (certainly an Oedipal thing going on here) but she persuades/seduces him into thinking that money, rather than love, is how to make their family happy. Thus, the story becomes less about the supernatural and more about Paul's odd quest to find happiness for his family. The trouble is that it is based on this materialistic goal.
In fact, the story insists upon this notion of morality being confused with money. Consider the last line of the story:
And even as he lay dead, his mother heard her brother's voice saying to her: "My God, Hester, you're eighty-odd thousand to the good and a poor devil of a son to the bad. But, poor devil, poor devil, he's best gone out of a life where he rides his rocking-horse to find a winner."
Note the repetition of words like "good," "bad," and "devil." Paul is forced to become something of a devil chasing money. Also note that Hester's brother is speaking of moral terms like "good" and "bad" in terms of money. "Eighty-odd thousand" 'to the good' and "a poor devil of a son to the bad." The terms of money and morality/ethics are confused. This is the point: that the mother confused money for happiness, confused it for what is actually good in life.