We also use the phrase "his luck ran out" to mean the same thing as people in the UK do. I think QM is right about the danger aspect in "lucked out." When I say it, I usually mean an implication that the situation could have been really bad, but it ended up being good. As in, "I almost got in a car accident, but lucked out since I slammed on the brakes in time." (Ok, that's an extreme example of a phrase that is used all the time, but it's a good demonstration). So, I got lucky and escaped a potentially bad situation.