Jakob Chychrun, Ryan White and Radim Vrbata scored for the Coyotes (21-31-7), who got 23 saves from Mike Smith.
Each team scored three goals in the first period on a combined 23 shots.
Schmaltz gave Chicago a 1-0 lead 37 seconds into the game with a one-timer off a feed from Toews. Chychrun tied it at 7:36.
Hartman put the Blackhawks up 2-1 at 10:28 with his 15th goal, and Kane made it 3-1 at 16:41. White and Vrbata scored 57 seconds apart in the final 2:18 to tie it 3-3.
Rozsival made it 4-3 with his first goal of the season at 2:56 of the second. Kane scored his second of the game at 12:41, and his third put the Blackhawks up 6-3 at 9:49 of the third.
"We got a couple goals late there [in the first]," Coyotes coach Dave Tippett said. "I don't think anyone expected it to be 3-3 after the first period, but both [teams] made a couple of mistakes in coverage early on. We kept making them and not getting saves, and the game ended up where it did. It was a poor game right from our goaltender out."
Chicago defenseman Niklas Hjalmarsson left in the first period with an upper-body injury and didn't return. He played five shifts.