But mostly what happened is that the puck stopped hitting the goalie.
tri answered this even more already but it's a lot to do with this, level of competition, and luck.
if we want to get specific, i think the devils played the 5 minute PK wrong anyway. in a case like that, when a team has SO much time to get something, the object should be making them waste as much as possible. once they got into the zone, the devils continued to chase. i'm not saying go to a wide passive box, but i would have looked to collapse (ala NYR) and let the kings spend 30 second clips passing around looking for a shot instead of having openings in high scoring areas like they did. also, if you're looking at it in the context of the game, the devils were likely in a position to take liberties clearing guys out around the net and slot because the refs wouldn't have called more.
if they are collapsed in that instance, you could argue the first two goals don't happen because brown would not have been able to get into the slot and carter wouldn't be able to be in that close to marty