Tyler Seguin and Martin Hanzal also scored for Dallas (1-2-0), which got coach Ken Hitchcock his 782nd career victory, tying him with Al Arbour for third most in NHL history.
Justin Abdelkader and Anthony Mantha scored for the Red Wings (2-1-0). Detroit's Mike Green had two assists.
Klingberg scored 9:15 into the game on a wrist shot from the high slot, a flick which struck the left goalpost and deflected off goalie Petr Mrazek's right skate before crossing the line.
The Stars took a 2-0 lead after Seguin scored his second of the season with Dallas on the power play 31 seconds remaining before the first intermission. Seguin one-timed a Klingberg pass over Mrazek's blocker, beating him top shelf and short side from the top edge of the left circle.
Dallas captain Jamie Benn got his 300th career assist on the goal.
Detroit answered 3:50 into the middle frame when Abdelkader scored with the Red Wings on a four-on-three power play. Abdelkader tapped the puck in at the near post following a cross-ice pass from Henrik Zetterberg.
Dallas regained its two-goal edge when Janmark scored his first goal of the season with a soft wrist shot that deflected off Mrazek's right pad before resting inside the near post.
Mantha made it a one-goal game when he beat Bishop far post with a wrist shot from the left circle with 4:49 remaining.
Mrazek stopped 29 of 32 shots in a losing effort for Detroit before leaving the ice with 3:00 remaining. Hanzal added an empty-netter with two seconds left.
Red Wings defenseman Danny DeKeyser sustained a lower-body injury late in the second period and did not return.
NOTES: The Red Wings scratched LW David Booth and D Niklas Kronwall (groin). ... The Stars scratched D Dan Hamhuis (groin), D Greg Pateryn and C Gemel Smith. ... Red Wings D Trevor Daley was playing in Dallas for the first time since spending 10 seasons with the Stars (2005-15). Daley was honored with a video tribute during the first period. ... Stars G Ben Bishop, who took a puck to the head Friday in the third period against Vegas and left the ice, returned to action after sitting out one game.