Smith was already having a good night before sending a wrist shot the length of the ice into the vacated Red Wings net. Smith made 31 saves before scoring his first NHL goal.
Mike Ribeiro scored twice in the third to help the Coyotes storm back from a 2-0 deficit in the opening frame. Martin Hanzal's goal at 7:08 of the third, in his 400th NHL game, put Phoenix ahead 3-2. That came 2:01 after Ribeiro had tied it.
His third of the season with 6:46 remaining gave Phoenix a 4-2 lead before Smith capped the win in dramatic fashion.