Detroit will open the NHL regular season in Stockholm with a two-game series against the St. Louis Blues on Friday and Saturday.
Zetterberg, a Swedish native who sat out the first eight preseason games in North America because of a sore groin, received the biggest ovation before the game from the record crowd of 8,398.
He responded with a dazzling shot into the top corner 15:08 in to give the Red Wings a 2-0 lead.
"It was cool," said Zetterberg, who added that his groin felt OK. "It was the first time the fans have cheered me so much in this arena. To come here and play in Sweden is fun, and it's extra fun to face a Swedish team."
Fellow Swede Niklas Kronwall, who had a goal and two assists, helped set up Zetterberg's score. Jason Williams, Valteri Filppula, Brian Rafalski and Todd Bertuzzi added goals for Detroit, which had a 30-24 shots advantage.
The game was played on the larger European ice surface, but the two-game series against St. Louis will be held on an NHL-sized rink.
Farjestad, which has been on the ice since August, won the Swedish championship last season and is currently third in the 12-team Elite League.