Fischer tied it 1-1 at 7:08 on a one-timer off a cross-crease pass from Stepan.
Strome scored with two seconds left on the power play to give the Coyotes a 2-1 lead at 12:23. Vinnie Hinostroza whiffed on a shot, but got control of the puck and passed in front to Strome, who redirected it into the net.
"Even though we were up 2-1 we didn't like our first, which is good," Tocchet said. "In the second, I thought we grabbed the game back."
Fischer scored 29 seconds into the second period to extend the lead to 3-1 on a redirection of a pass from Stepan in front.
"They checked well," Blue Jackets forward Boone Jenner said. "You could see it throughout the second period, they had good sticks."
Fischer completed the hat trick at 4:44 of third period to make it 4-1 with a shot from the high slot that beat Bobrovsky glove side.
"[Brendan Perlini] drove the goal on the second and third ones," Fischer said. "That's goes unnoticed by people watching.
"A lot of credit to [Stepan] and Brendan. [Stepan] is the anchor on the line. He gives us confidence knowing we're going to get the puck."
Coyotes forward Alex Galchenyuk, who was acquired in a trade with the Montreal Canadiens for Max Domi on June 15, had one shot in his Arizona debut after missing the first seven games with a lower-body injury.
Blue Jackets defenseman Seth Jones led all Columbus skaters with 27:24 of ice time in his season debut after missing the first seven games with a Grade 2 MCL sprain in his right knee.
"I thought I made some good offensive plays, some good defensive plays," Jones said. "I don't know how much I played, but I felt pretty good out there."
