The Flyers continued to pressure with Elliott on the bench, and Sean Couturier scored his second goal of the game with 16 seconds left to tie it 3-3.
Two nights after allowing two third-period goals in a 4-3 loss at the New Jersey Devils, the Coyotes were staring at another potential excruciating loss.
"I wanted to stay calm," Tocchet said. "Obviously, you're upset because you can't believe that just happened. But we still have some pretty good players for 3-on-3."
After killing a hooking penalty against Jakub Voracek in overtime, the Flyers (6-5-1) allowed an odd-man rush. Max Domi got the puck to rookie Clayton Keller, who passed to Goligoski for a one-timer past Elliott for his first goal of the season.
"It just feels good to get a win," Goligoski said.
Keller extended his point streak to six games with the assist on the game-winner. He has four goals and four assists in that span.
"The whole building thought he was going to shoot and he made a nice play to Alex," said Tocchet, who got his first win as Arizona coach in the city where he started his NHL career and spent 11 seasons as a player. "He's a big-time player and he's 19 years old."
Jordan Martinook (2:20) and Christian Dvorak (7:46) each scored his first goal of the season in the first period to put the Coyotes ahead 2-0, and Brendan Perlini scored a power-play goal at 1:34 of the third period to make it 3-0.