The 25-year-old Klingberg leads NHL defensemen with 18 points.
''I see a young guy maturing,'' coach Ken Hitchcock said. ''He's just got a calmness to the way he's moving the puck, and it's helping, not only helping him, but it's helping his partner too.''
Klingberg's partner, Esa Lindell, had a goal and an assist.
''I just feel like I'm playing a really good all-around game right now,'' Klingberg said. ''Felt really good. The confidence is good right now.''
Bishop made 14 saves in his first shutout of the season. He got some help 3 1/2 minutes into the second period when Tyler Seguin swept the puck away from the net to prevent an Islanders goal.
''Ben was really competitive and he was battling like crazy in the net,'' Hitchcock said.
Klingberg got each of his points in the first. He put Dallas in front with his fourth goal 1:32 into the first, and then assisted on scores by Gemel Smith and Lindell.
Mattias Janmark added Dallas' first short-handed goal of the season at 6:54 of the second, and Jason Spezza made it 5-0 with a power-play goal with 4 minutes left in the period. Spezza ended a 16-game goal drought.
The Stars seemed to benefit from some changes to their lines. With Seguin now centering the Stars' second group, Smith was moved up to the top line. Jamie Benn has moved to center with Alexander Radulov on the right wing.
''It was a good feeling playing with that line, playing with two magicians,'' Smith said.
Jaroslav Halak had 24 saves for the Islanders, who have allowed a league-high seven short-handed goals.
''Some heart would be nice to see,'' Islanders coach Doug Weight told MSG Network after the game, ''some work ethic, maybe following a game plan and maybe some consistency as far as getting pucks in.''
NOTES: Klingberg, Benn and Seguin entered the game tied for the Stars' scoring lead with 15 points apiece. ... Radulov extended his points streak to seven games with his assist on Smith's goal. ... New York rookie Mathew Barzal had a six-game points streak end. ... The Islanders' Anthony Beauvillier was helped off the ice in the third period after a shot by Klingberg hit the defenseman's right ankle. ''You hate to see it,'' Weight said. ''It was the same exact spot twice. Hopefully he's OK.'' ... Dallas activated RW Brett Ritchie from injured reserve and placed Tyler Pitlick on IR. ... C Alan Quine returned from a conditioning assignment in the AHL to play his first game this season for the Islanders.
Islanders: Complete a two-night, two-game trip on Saturday at St. Louis.
Stars: Begin a three-game trip on Monday at Carolina.
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