Smith was playing the puck behind the net when Shaw barreled into him in the second period. Smith lay motionless on the ice for several minutes, but finished the game.
Shaw was assessed a five-minute major and game misconduct.
"The contact Shaw makes with Smith is not incidental," NHL disciplinarian Brendan Shanahan said. "Rather, it is a forceful blow delivered by Shaw's shoulder to Smith's chin. Also, in our opinion, Shaw does not make a reasonable effort to avoid such contact."
A source familiar with the situation told ESPNChicago.com's Jesse Rogers that the severity of the suspension was due to a goalie being involved. General managers have asked the league to crack down on incidents involving goaltenders.
The timing of Shaw's hit also played a role, according to another source familiar with the situation. The league has been receiving criticism since Nashville's Shea Weber slammed Detroit's Henrik Zetterberg's head into the boards at the end of Game 1 of their playoff series. Weber was fined but not suspended and the overwhelming response outside of Nashville was outrage.
The light punishment on Weber contributed to Shaw being made an example of, especially considering the victim of the hit was a goalie, according to the source. The league felt it could not let Shaw off with a fine, sending a poor message for the rest of the playoffs.
Smith, who didn't practice Monday but participated in the team skate Tuesday morning, was in the starting lineup Tuesday night as Phoenix won 3-2 in overtime.
The Coyotes lead the series 2-1. Game 4 is Thursday night at the United Center.