Man down/up posession in overtime

I was recently at a game where the score was tied. Team A had a 1 minute personal foul called on them with about 30 seconds left in the game. Team B's manup unit took a shot with a few seconds left, missed, and Team A's goaltender "backed up" the shot and was awarded the ball with about 2 seconds left. The whistle blew, and regulation time expired with Team A's goalie holding onto the ball.
At the beginning of the first overtime there was a faceoff.
My question: is there something special about overtime which causes a face off in this situation instead of Team A starting with the ball?
At the beginning of the first overtime there was a faceoff.
My question: is there something special about overtime which causes a face off in this situation instead of Team A starting with the ball?