"If you can score some timely goals on the power play and kill some timely penalties you can win a lot of games," Zajac said.
The Devils (3-2-1) finished 2-for-3 on the power play. The Coyotes finished 1-for-3.
Hall had two goals and an assist, Adam Henrique scored, Pavel Zacha had two assists, and Cory Schneider made 28 saves for the Devils, who won their second straight game.
The win was the third straight at home for the Devils. The last time New Jersey began the season with three straight home wins was 2002-03.
"We want to establish that this is our home rink and teams coming in here will have a tough night," Zajac said. "Win or lose, they'll have to work for everything."
Defenseman Oliver Ekman-Larsson had two goals, Jordan Martinook scored, and Justin Peters, who was making his first start of the season, made 30 saves for the Coyotes (1-5-0). Arizona, playing the fifth of a six-game Eastern Conference road trip, has lost five straight after a season-opening 4-3 home victory against the Philadelphia Flyers on Oct. 15.