In Game One, a Rangers turnover was the cause of Justin Williams' game-winning goal. The Game 2 OT winner was the result of some good circumstances for the Los Angeles Kings, but there was no scapegoat on the New York Rangers. Rather, Dustin Brown simply made the most skillful play he's made all playoffs, at a very important time.
Let's deal with the elephant in the room first: the faceoff took place in the defensive zone, after a clearing attempt by the Kings went over the glass. The Rangers thought there should have been a penalty on the play. The Kings thought it deflected off the glass before it went out of play. I haven't seen a video of the incident, and the opinion of the Internet appears to be mixed (though I should probably find some reactions that aren't from New York or LA). The Kings not being on a PK obviously was important, but this goal was not a direct result of that call.
Willie Mitchell dumps the puck as soon as he gets to the red line, because these are the Kings and that is Willie Mitchell. The Rangers play this pretty well, and with Ryan McDonagh getting to the puck first, they had an opportunity to escape their defensive zone despite LA getting their forecheckers in deep.
There's a problem for the Rangers, though. To use a soccer phrase, McDonagh's first touch was too heavy; in normal terms, he couldn't control it, and it deflected back the other way off of his stick. Dustin Brown doesn't slow down for anything, so he was going at full speed and ended up grabbing the loose puck before McDonagh could recover.
St. Louis has since gotten back into the play, but all three forwards are down low along with the two defensemen. So, Brown goes up to Willie Mitchell, who's got a lot of room. Mitchell decides to take his time to find a shooting lane instead of shooting right away. So instead of facing this:
He's facing this:
Mitchell's shot only makes it as far as that cluster of players in front. Fortunately, one of the players in that cluster is Anze Kopitar, and he reacts quickest to get to the puck.
So what happens? Again, the Rangers' defense and stays low, this time preventing Kopitar from getting the puck to Brown in front. When Kopitar eventually spins around and feeds the puck back to Mitchell, Martin St. Louis is only able to get within five feet of Mitchell, and Mitchell one-times it instead of waiting for traffic or a shooting lane. You may also notice Brown fighting for position in front of the crease; he eventually forces his way in front of McDonagh.
That makes a big difference when Mitchell takes his shot. As you can see, there's essentially no one in between Brown and Mitchell, and even though the shot is heading wide of Henrik Lundqvist, Mitchell has given Brown a chance to make a play on it. And does he ever.
KINGS GOAL: Dustin Brown (5) Tip-in - ASST: Willie Mitchell (3), Anze Kopitar (20) - 10:26, 2OT