Defensemen Bryan Allen and Jamie McBain scored for the Hurricanes, who posted their first win in this series since January 13, 2006. Cam Ward made 26 saves in his return from missing three games with a lower-body injury.
Carolina improved to 2-0-2 on its six-game homestand.
Gabriel Bourque had a goal and an assist for Nashville, which had earned at least a point in its previous five games (4-0-1).
Seldom-used backup Anders Lindback stopped 29 shots in the loss, his sixth straight. He made just his second appearance since January 17.
The Hurricanes trailed 1-0 and 2-1, but a pair of power-play goals gave them the lead.
It was tied 1-1 5:17 into the second period, as McBain fired a slap shot past Lindback from the right point. But Nashville moved back in front at 12:31 when Colin Wilson stripped McBain in the Carolina zone and buried his own rebound.
Thirty-three seconds later, the game was tied again.
Newcomer Andrei Kostitsyn, who the Predators acquired from Montreal prior to Monday's trade deadline, committed a hooking penalty, and the ex-Canadien observed from the box as Staal took an indirect pass off Jeff Skinner's chest and fired the puck into an open net.
Tim Brent then scored on a scramble in the crease with 2:36 remaining in the middle stanza.
Nashville opened the third period with a man-advantage, but Allen scored on a breakaway -- catching up to Staal's pass from the defensive zone -- after exiting the penalty box at 1:28. It was his first goal since January 26, 2011, snapping a 94-game drought.
Bourque converted his own rebound through a prone Ward 50 seconds later to draw the Predators within 4-3.
Ward finished with 15 saves over the final 20 minutes.
Lindback stopped 13 shots in a first period that ended with the Predators in front on Craig Smith's early tally. The rookie center knocked down Ward's clearing attempt inside the blue line, leading to a wraparound bid by Matt Halischuk that was foiled. Smith collected the loose puck at the right post and chipped it home at 2:42.
Staal now has seven goals and nine assists during a nine-game point streak...Carolina now has scored a power-play goal in four straight games and five in a row on home ice...The Predators had not allowed a power-play goal in seven consecutive games.