Fleury made a pad save against Jason Robertson from the low slot on a 3-on-2 at 17:25 of the first period. He then denied Roope Hintz from point-blank range with 6.5 seconds remaining in the second period.
"I thought there was only one or two seconds left so I was kind of … I shot it too fast," Hintz said. "If I knew there were six seconds, I would hold it a little bit more and wait for his glove to drop. It is what it is. That was a nice save, and it didn't go in.
"He's a great goalie. He had me today, but I will get him next time."
Fleury chalked it up to good fortune.
"Sometimes I think I'm lucky," he said. "That's what I meant to do, right? I'm hoping to stop it, obviously. As a goalie, those are the ones I play for and I love doing, you know. Those are the ones that feel good inside. Players like to score. And for me, it's like I make a save that's a little bit (of a) tough one. Those are the ones that feel good. Makes me smile."
Tyler Seguin almost won it for the Stars 36 seconds into overtime, but his diving attempt hit off the crossbar.
Oettinger made his best saves shortly after to help get the game to a shootout. He stopped Patrick Kane on a 2-on-1 at 45 seconds before getting across to make the save on a one-timer from Alex DeBrincat 10 seconds later.
"He's on his game right now, and we need him," Dallas coach Rick Bowness said. "He's making huge saves for us. We played really well defensively, and so did they. That was probably the best defensive game they played in a while. I'm sure (Chicago coach) Derek (King) got after them after giving up seven last night. And Fleury was great for them. He made some phenomenal saves for them, but Jake made the big saves at the right time."
King said he was happy with his team's effort.
"It was good," he said. "Solid effort and that's a good hockey team. Obviously, 'Flower' made that big save for us, but for the most part we were pretty solid."