Klingberg, who went 22 games without a goal, won it after Blackhawks forward Patrick Kane took a hooking penalty at 1:31 of overtime. The Stars defenseman took a cross-ice pass from Joe Pavelski and scored with a wrist shot from the right face-off circle.
"My shot the last few games, I told you I was going to shoot more," Klingberg said. "I had my chances today, so it was nice to see it go in, and obviously a great pass by [Pavelski]. I literally had an open net, so it was nice."
Pavelski had two goals and two assists for the Stars (14-12-2), who were 3-for-8 on the power play. Jake Oettinger made 20 saves.
Dominik Kubalik tied it in the third period for the Blackhawks (11-15-4), who trailed 3-1 in the second. Kevin Lankinen made 35 saves.
"A little bit of adversity. We had to climb back," Chicago forward Dylan Strome said. "I thought we did a good job of that. Coach (Derek King) came in and said, 'adversity comes in all different shapes and sizes, big or small,' and that was obviously big.
"We did a great job of clawing back and sticking with our game and found a way to get a point. I think we'll take the positives."