Tortorella made the announcement at the team's summer summit for fans.
Gulutzan was fired as Dallas Stars coach May 14 after going 64-57-9 in two seasons.
"He brings youth, he brings the desire," Tortorella said of Gulutzan, 41. "He's going to be one [heck] of a coach in this League -- as a head coach -- and we are very fortunate to have him on our staff."
"He's just such a good teacher," Tortorella said. "I think that's a very important part of the game, is the teaching part of it."
"I'm really excited about the staff and how we think, how we think the game and how we're going to go about the business," Tortorella said.