Brind'Amour scored both of his goals during a five-goal first period for Carolina. It was the 39-year-old center's first multigoal game of the season.
Steven Stamkos scored his team-leading 47th goal, keeping him one behind NHL leaders Alex Ovechkin and Sidney Crosby.
Bowman scored the first two goals of his career in the first period and Jussi Jokinen added another one to chase Lightning goalie Antero Nittymaki. Carolina had 23 shots in the period.
Brandon Bochenski scored twice for the Lightning and Andrej Meszaros had a goal and an assist for Tampa Bay.
Niitymaki allowed five goals on 23 first-period shots.
Staal scored both of his goals in the second period, the latter on a breakaway shot past backup goalie Mike Smith to give the Hurricanes a 7-2 lead. He assisted on a goal by Jussi Jokinen and an empty-net goal by Chad Larose to end the game.
NOTES: Carolina goalie Manny Legace skated off the ice midway through the second period and is listed day-to-day. ... Lightning C Zenon Konopka had two fights and set the team record for penalty minutes in a season with 260