Staal scored at 4:19 of the second period to give Carolina a 2-1 lead. After Staal won a face-off to Pesce, the Hurricanes captain took a return pass and split Predators defensemen Mattias Ekholm and Roman Josi before scoring top shelf near the goal line.
"I pretty much lost the puck a little bit going between those two guys," Staal said. "When it goes well is usually when you're not really thinking. I was just kind of finding a way to get to the net. It was a good goal."
Haula tied it 2-2 at 8:41 when he took a pass from Matt Duchene and scored when his shot went in off the back of the sweater of defenseman Jani Hakanpää at the right hash marks.
Nashville defenseman Alexandre Carrier and forwards Tanner Jeannot, Yakov Trenin and Mathieu Olivier each made his NHL playoff debut.
"We have a lot of guys who have made cases for themselves to play, and obviously the group tonight didn't get it done," Hynes said. "We have some guys who are not in the lineup that we know are ready to play and can really contribute, so that's all part of the process."
Teams that win Game 1 are 490-222 (68.8 percent) winning a best-of-7 NHL playoff series, including 7-1 in the first round last season.
NOTES: Defenseman Jaccob Slavin was plus-1 with four hits in 21:25 after missing Carolina's final regular-season game, a 5-0 loss at Nashville on May 10, with a lower-body injury. ... Nedeljkovic is the first rookie goalie to win a playoff game for the Hurricanes since Cam Ward in Game 7 of the 2006 Stanley Cup Final against the Edmonton Oilers. ... Forward Steven Lorentz had an assist on Teravainen's goal in his NHL playoff debut. … The Hurricanes outhit the Predators 56-49. … Josi was minus-3 with one shot on goal in 24:04. … Haula played 41 games for Carolina last season, scoring 22 points (12 goals, 10 assists), before being traded to the Florida Panthers on Feb. 24 with three other players for Trocheck.