The Devils, winning their second in as many nights, finished with a 49-15 edge in shots but trailed 1-0 until tying it when Patrik Elias' seemingly harmless, no-look pass from along the boards deflected off Penguins defenseman Hal Gill's right skate and into the net with 2:29 remaining in the third.
Fleury played an exceptional game, making key save after key save as the Devils held a 20-2 edge in shots in the third period and a 35-6 advantage in the second and third periods.
Despite going long stretches without seeing a shot, Devils goalie Martin Brodeur - required to make only 15 saves - got his 540th career victory, 11 short of Patrick Roy's NHL career record.
Parise and Elias also had the only Devils goals in a 2-1 win over the Islanders on Friday night.
The Penguins played their first game at home, and in North America, since opening the season by splitting two games with Ottawa in Stockholm last weekend. There were new banners hanging from Mellon Arena's roof commemorating the division and conference championships they won last season en route to their Stanley Cup finals loss to Detroit, but there was no pre-game banner raising ceremony.
There weren't many goals, either, with Miroslav Satan - picked up from the Islanders after last season - scoring his first goal for Pittsburgh on a power play midway through the first period.
The Penguins, one of the NHL's top-scoring teams a season ago, couldn't find much offense after that, a continuation of their inability to generate many scoring chances during their 3-1 loss to Ottawa in the second game of the Sweden series.
Fleury's best save might have been with Brooks Orpik off for hooking in the third. Jamie Langenbrunner's backhand toward an open net hit a post and deflected to Parise with Fleury on his back, but Fleury stretched out his left arm and knocked down Parise's bad-angle shot.
A few minutes later, Brian Gionta stole a Sidney Crosby pass near the Penguins' blue line and had a short breakaway, but the goalie stopped Gionta's slap shot.
Notes: The Devils, beaten out for the Atlantic Division title by the Penguins last season, have won 16 of their past 21 in Pittsburgh. ... Crosby hasn't scored in their first three games.
His two grandmothers from Nova Scotia, Linda Crosby and Catherine Forbes, attended their first game in Pittsburgh. ... The Penguins have gotten at least one point during 19 of their past 22 home openers.
