On November 13th, 1934 St. Louis Eagles' defenseman Ralph "Scotty" Bowman (no, not that Scotty Bowman) scored the first penalty shot goal in NHL history.
It came during a 2-1 Maroons win over the St. Louis Eagles. Alec Connell was the Montreal goaltender. The goals was also Bowman's first NHL goal, and just the second penalty shot attempt in NHL history.
Just three days earlier, on November 10th, the first NHL penalty shot was awarded. However Montreal Canadiens forward Armand Mondou was unable to score on Toronto Maple Leafs goalie George Hainsworth.