Kevyn Adams and Erik Cole also scored for the Hurricanes, who led almost the entire way in a matchup of the past two Stanley Cup champions. Rod Brind'Amour added three assists, while Cam Ward finished with 19 saves.
Vincent Lecavalier and Vaclav Prospal each had a goal and an assist for the Lightning, while Ruslan Fedotenko and Martin St. Louis also scored.
The Hurricanes took control early, outshooting the Lightning 13-2 in the first period and chasing Marc Denis early in the second after scoring four goals on 17 shots en route to a season-high scoring output. And it all began with that record-setting flurry in the first 20 minutes.
On the first goal, Staal took a pass along the boards from Justin Williams and beat Denis from the right faceoff circle on the power play to make it 1-1 at 8:23 of the period.
On the ensuing faceoff, Dan Boyle whiffed on a pass in his own end, sending the puck directly to Craig Adams. Denis stopped Adams in close, but Kevyn Adams got to the rebound and pushed it into the net just 9 seconds later.
Then, with the Hurricanes pushing up ice, Cole dropped a pass to Ray Whitney on the right side, then got the puck back at the left post for a give-and-go goal at 9:06 to make it 3-1.
The three-goal burst beat the previous franchise record of 63 seconds set by the Hartford Whalers against Toronto in January 1987. The NHL record for fastest three goals is 20 seconds by the Boston Bruins against Vancouver on Feb. 25, 1971.
It also fit right in with what has already been a wild season series. The Hurricanes and Lightning had split their meetings by identical scores, with Williams notching his first career hat trick in Carolina's 5-1 win on Oct. 16, while St. Louis scored three goals in 6:17 of the second period in Thursday's victory.
Staal and Walker came close to their own hat tricks. Staal pushed the lead to 4-1 on a second-period power play, scoring when his pass for Cole at the top of the crease ricocheted off Paul Ranger's skate and through Denis' legs at 5:14.
The Lightning closed the gap to 4-3 on Lecavalier's power-play goal late in the second. But the Hurricanes answered less than a minute later when Staal found Walker - who had a step on Doug Janik as he crossed the blue line - and Walker beat Johan Holmqvist at 17:34 of the second and answer Tampa Bay's rally.
Notes: Walker also scored on a rebound of a shot by Whitney early in the third. ... Staal pushed his goal streak to three games. ... Tampa Bay has been outscored 14-5 in the opening period this year. ... Carolina had won just five of the previous 21 meetings. ... Lecavalier, Fedotenko and St. Louis each pushed their point streak to four games.