Palmieri and Zajac scored two goals each to lead the Devils to a 5-2 victory Saturday night in the NHL Global Series game at Scandinavium.
"We feel like we have a good group, we have a strong culture," New Jersey coach John Hynes said. "We like we have an identity of how we want to play, the types of players we want, how we want to be off the ice. Those are really the things that matter. We want be a tough team to play against."
Stefan Noesen also scored, Miles Wood had two assists, and Keith Kinkaid made 17 saves.
"We played a pretty good hockey game from the start," Zajac said. "Not too many lapses, in our play throughout the game, our speed eventually kind of took over. We played the right way, especially with the lead. It’s early in the season and it’s a good sign."
Milan Lucic and Leon Draisaitl scored for the Oilers.
In the new season, Palmieri didn’t waste time.
He picked up the puck near the boards in the offensive zone, moved unchecked toward the net from the left and beat Cam Talbot inside the far post on the first shot on goal still in the opening minute.
The Oilers came back on a power play, with captain Connor McDavid setting it up. The star centre used his speed to carry the puck up ice from his own up zone and skated around the goal to feed Lucic in front for a one-timer to tie it at 4:01.
New Jersey took control early in the middle period with goals from Palmieri and Zajac in a 1:09 span. Palmieri scored on a power play for his second, and Zajac doubled the advantage after Talbot lost the puck behind the goal. The Devils clearly dominated at this stage, outshooting the Oilers 10-4 in the period.
"Obviously, we weren’t good enough, they were good," McDavid said. "We did some good stuff, we did some bad stuff too."
Zajac added his second in the third after a defensive blunder by defenceman Matt Benning before Draisaitl reduced the deficit to 4-2. Noesen finished it off with an empty-netter.
"Give them credit, they pressured us in areas that give us a lot of headache and we weren’t able to do the same for them," Edmonton coach Todd McLellan said. "We had a tough night, it was evident," he said. "Our puck move, our puck support, our ability to play fast and clean was not very good."
It was the 23rd NHL regular-season game played in Europe, the 11th in Sweden and the first in Gothenburg.
Prior to Saturday, the Devils completed a training camp in Switzerland where they played SC Bern in an exhibition game while the Oilers came from practice in Germany where they faced Cologne in an exhibition.
The NHL will return to Europe next month. On Nov. 1-2, the Florida Panthers and the Winnipeg Jets will play each other in a couple of regular season games in the Finnish capital of Helsinki.
It was only the second time in NHL history that the last two Hart Trophy winners faced each other in a season-opening game opener. It was also only third time that the reigning Art Ross Trophy winner (McDavid) and reigning Hart Memorial Trophy winner (Taylor) played against each other in a season opener.
Oilers: At Boston on Thursday night.
Devils: Host Stanley Cup champion Washington on Thursday night.