I've beaten people by .01 in Mario Kart 64 before, but I consider my luckiest moment to be when I got that power star in the top of the clock in Super Mario 64.
I'm not an avid gamer, so I'm naturally sort of clumsy with a control stick. I had tried to get this thing since 1997 and would try every few months or so. Didn't get it until April 2006. But when I did get it (it was the 120th one) I was so happy. Yeah, this just shows that I'm awful at even the easiest of games, but for a guy who rarely plays a game aside from that two month stretch in 1997 (it rained almost every day so I played it a lot that summer) I was pretty happy.
Mario Kart seems to be everyone's "lucky" game here so here's another story. I was cruising along fine once and then disaster struck. A lightning bolt, a blue shell, another lightning bolt (on the stadium so I fell down), and then a red shell by an opponent who was so far behind and wound up beating me. Yes, it didn't work out for me, but great luck on my opponent's part! Too bad I wanted a rematch, being the really competitive person I am (it develops from sports, trust me).