"There's an upbeat tempo to our game," Brodeur said. "We have skill. I think we have the firepower to play a certain way, compared to past years, that's for sure."
Parise's slap shot from the slot with 5:46 left in the first gave the Devils a 2-1 lead.
Rolston and Elias scored even-strength goals 4 1/2 minutes apart in the second period to push the Devils' lead to 4-1.
"We played well, made some mistakes, but we got the win and that's the most important thing," Devils coach John MacLean said.
Moulson scored at 14:28 of the second period to pull the Islanders to 4-2, and his second goal closed the scoring at 11:18 of the third.
"I hope I score two goals every game," said Moulson, who had a career-high 30 goals last season. "I think I have a lot to prove. I want to score goals and I want to win."
Joensuu beat Brodeur with a backhander for the game's first goal at 1:38 of the first period. Arnott tied the score 3:17 later, tipping in a shot by Andy Greene past Roloson.