Alfredsson, Ottawa's captain who holds the Senators' career mark in games played (854), goals (331), assists (516) and points (847), was held pointless as he returned to his native Sweden for the first time with his NHL team.
Working out hard during the offseason paid off for Kennedy.
"I think I worked hard in the summer and drew confidence out of that," Kennedy said. "So I came into the season with confidence."
Kennedy, who also scored 40 seconds into the first period in front of 13,699 at a soldout Globe Arena, beat Ottawa goalie Martin Gerber with a wrister from the left circle.
The goal came after Spezza turned over the puck to Kennedy at the blue line. Kennedy skated in all alone and scored.
"I'm glad to get it off my back for one, and for the second I'm just excited to help the team win and get the season on a roll here," Kennedy said.
"He (Spezza) just kicked the puck up and I just tried to get the puck towards the net and shoot as hard as I can. I try to get as many pucks on the goal as I can, and tonight they went in."
Marc-Andre Fleury made 32 saves for the Penguins. Gerber stopped 26 shots for the Senators.
The teams will meet again on Sunday.
Defenseman Rob Scuderi forced overtime when he beat Gerber with a wrister from the left circle at 3:56 in the third period.
Sidney Crosby, who finished the 2006-07 season as the first teenager in NHL history to win the scoring title, had the assist.
Evgeni Malkin made it 2-1 for Pittsburgh with a short-handed goal 3:18 into the second period. The Russian center stole the puck from Christoph Schubert in mid-ice, skated in all alone and swept a backhand past Gerber.
Heatley, unchecked between the circles, tied it 2-2 with a power-play goal at 12:15. Spezza gave Ottawa the lead for the first time with a short-handed goal at 17:55, beating Fleury with a wrister after a quick 3-on-2 breakaway.
Pittsburgh got off to a strong start when Gerber gave up a goal in the opening minute. Kennedy skated in from behind the net and wristed the puck from a tough angle past Gerber.
Later in the period, Kennedy nearly made it 2-0 when he tipped a pass just wide in front of Gerber.
Pittsburgh's power play was sloppy in the opening period, failing to capitalize on five straight chances with the man advantage.
Crosby agreed that Pittsburgh needs to improve its power play.
"They've been giving a lot of teams trouble with their penalty killing," he said of Ottawa "You have to be patient against them. There's no doubt we need to be better.
"We've had a successful power play in the past. It takes practice and it takes execution."
The NHL is returning to Europe this weekend with regular-season openers in both the Swedish and Czech capitals. In Prague, the New York Rangers beat the Tampa Bay Lightning 2-1.
NHL players in Sweden is nothing new. Canada played an exhibition in an adjacent arena to the Globe before heading to Moscow for the last four games in the historic Summit Series in 1972 that had NHL players against the Soviet Union for the first time.
And many NHL teams, among them Original Six clubs like the Montreal Canadiens, New York Rangers and Toronto Maple Leafs, and clubs from the old World Hockey Association have played tournaments in Sweden throughout the years.
But this was the first time in league history two teams began the regular season in Sweden, one of Europe's top hockey nations.
The teams last met in last season's playoffs when the Pittsburgh swept Ottawa in four games in the first round.
This was the second time Pittsburgh played the first two games of the regular season on international ice. The Penguins traveled to Tokyo in 2000-01 splitting the two-game series with the Nashville Predators.
Ottawa was listed as the home team, but the Senators drew several boos when they entered the arena despite having Alfredsson on the team.
Mats Sundin, who helped Sweden win the 2006 Olympic gold medal and a Stockholm native, drew the biggest cheers when he dropped the puck during a ceremonial faceoff before the game.
Sundin, an eight-time All-Star with the Toronto Maple Leafs, has not yet signed a contract with an NHL club this season.