St. Louis got the game-winner from in-close. The Tampa Bay captain also had an assist and moved past Neal Broten (923) for 93rd place on the career points list with 924.
Steven Stamkos scored a tiebreaking power-play goal as the Tampa Bay went up 5-4 at 17:31 of the third.
Chicago's Jonathan Toews was credited a game-tying goal with 1:49 left in the period after the Lightning put the puck into their own net.
Chicago took a 4-3 lead 93 seconds into the third on Shaw's in-close power-play goal. It was the first time the Blackhawks have scored during the third period since opening night.
Hedman's first goal in 41 games tied it at 4 at 11:03 of the third.
After blowing a two-goal lead, Stamkos gave Tampa Bay a 3-2 advantage at 17:21 of the second when he beat backup goalie Nikolai Khabibulin from the top of the right circle. It was just Khabibulin's second start in Tampa since helping the Lightning win the 2004 Stanley Cup.
Kane pulled the Blackhawks even at 3-3 on a breakaway goal with 1:07 remaining in the second.
Bickell cut the Blackhawks' deficit to 2-1 with a rebound goal with 11 minutes left in the second. The left wing has a goal in three consecutive games.
Tampa Bay took advantage of its first power play as Carle put the Lightning up 1-0 6:28 into the game. The Blackhawks entered with the NHL's third-worst short-handed unit.
Thompson stopped a 19-game goal drought to make it 2-0 during a short-handed 2-on-1 at 1:24 of the second.
NOTES: There were a number of Hockey Hall of Famers with ties to the Blackhawks in attendance. Current Lightning broadcaster Phil Esposito spent time with former teammate Bobby Hull. Others at the game included Stan Mikita and current Blackhawks senior adviser Scotty Bowman. ... Stamkos has nine goals and 14 points in seven games against the Blackhawks. ... Chicago D Michael Kostka is expected to miss four to six weeks due to a lower-body injury. He was hurt during practice Tuesday.