Klingberg scored only three goals in his first 36 games this season, but has scored three in the last five.
"With the team playing better, the confidence has grown a lot for me," he said. "With a lot of injuries at the start of the season, I think I put a lot of pressure on myself to be the guy you can really count on offensively, and it hurt maybe my overall game a lot."
Klingberg scored 21 goals in his first two NHL seasons.
"I think defensively I've been playing really good. Offensively, I've been getting a lot of chances."
"We created enough chances in the third to score at least two," Zetterberg said, "but it's not going our way right now. Just got to keep battling."
Niemi finished with 31 saves, including 16 during a pressure-packed third period when the Stars were protecting the 4-2 lead Klingberg and Johns provided.
"He played fabulous," Dallas coach Lindy Ruff said. "He made three or four highlight-reel saves for us."
Red Wings coach Jeff Blashill seemed frustrated because Detroit outshot Dallas 33-21 but still lost.
"Sometimes goalies play real good, sometimes the puck doesn't go in," he said. "That's why you can't give up four, so we got to be tighter in that for sure."
Patrick Eaves' empty-net goal with 1:06 to play relieved that pressure.
But the biggest goal came at 15:53 of the second period, when Klingberg put in a rebound for a 3-2 lead.
"That's a play that he just sort of sniffed out and got in there as the fourth guy, and found the puck," Ruff said. "At that time of the game, that was a huge goal for us."
Detroit captain Zetterberg saw the play from a different angle.
"It was a big save by Petr (Mrazek) first of all, and the puck was just stuck there," he said. "Tough to know if you want to play it or not play it. It ended up on their stick."
Brett Ritchie had tied it at 2 on a tip-in goal at 3:31 of the second period.
Johns sent a wrist shot from the right-wing boards off defenseman Jonathan Ericsson's stick and past Mrazek at 16:42.
Dallas' Lauri Korpikoski scored the game's first goal.
The Red Wings completed a seven-game trip with a 2-3-2 record, and remained last in the Atlantic Division.
Detroit had won the previous seven games at Dallas dating to Oct. 14, 2010.
Detroit's goals came 1:33 apart late in the first period.
With a 5-on-3 advantage, Nyquist took a cross-ice pass from Thomas Vanek for an easy wrist shot from just to the right of the net with Niemi pulled in the other direction.
The power-play goal was only the Red Wings' third on the road this season. The other two were against the Rangers on Oct. 19. Detroit has the NHL's worst power-play success rate at 11 percent. The Stars' penalty killers entered the game 29th in the league (74 percent).
Stars LW Jamie Benn returned after missing four games because of a foot injury. ... In 43 games against Dallas, Zetterberg has 47 points (12 goals, 35 assists). ... Korpikoski has four points (two goals, two assists) in the last two games. ... Niemi is 14-3-3 against Detroit in his career. ... Vanek has 10 points (five goals, five assists) in his last nine games. ... Mrazek had 16 saves.
Red Wings: Return home to play against Pittsburgh on Saturday.
Stars: Complete a two-game homestand on Saturday against Minnesota.