Smith, 55, joined the Coyotes in August 2005 and spent the last two seasons behind the bench with head coach Wayne Gretzky.
"I will continue to support Wayne and the Coyotes organization and help them in any way I can," said Smith. "I would also like to thank all the Coyotes players and training staff that I coached and worked with over the past two seasons.
"It was fun and rewarding to watch our players' progress."
Gretzky referred to Smith as "a great coach and a true professional."
"This is a great opportunity for him to be a head coach and I know he will be successful with SKA St. Petersburg," said Gretzky. "We will miss him as a coach and, more importantly, I will miss his companionship.
"I wish him and his family good luck."
Smith spent 11 seasons as an associate coach with the Detroit Red Wings prior to joining the Coyotes. In his NHL career, he has contributed to five Stanley Cup championships: three with Detroit (1997, 1998 and 2002) and two with Pittsburgh (1991 and 1992). He previously spent three years as assistant coach of the Buffalo Sabres.
Smith will serve as an associate coach for the U.S. team at the world championships in Moscow, April 27-May 13. He was an assistant to Bob Johnson for the 1991 Canada Cup and to Ron Wilson at the 2004 World Cup of Hockey.
Smith has also coached in Sweden, Norway and Italy.