"Anything can happen in overtime," Chicago coach Joel Quenneville said.
Chicago was coming off a 3-2 shootout win Thursday night at Florida.
Stamkos' final goal, a left circle power-play shot with 32.2 seconds remaining in the second, tied it at 4. The center had pulled Tampa Bay within one at 11:45.
"An up-and-down period," St. Louis said. "We take the lead 2-1, and (all) of a sudden bang, bang, bang we're down 4-2. We keep pushing and got back our game a little bit and get rewarded. There's no such thing as a bad win in this league."
After Tampa Bay went up 2-1 on Connolly's goal at 4:55 of the second, the Blackhawks responded with three scores from Hossa (6:17), Carcillo (7:10) and Sharp (7:22) over a 1-minute, 5-second stretch.
Hossa scored from the low left circle, Carcillo stopped a personal 28-game goal drought during a breakaway and Sharp beat Dwayne Roloson from the slot.
"You're never happy after a loss," Blackhawks goalie Ray Emery said. "It's frustrating to have a 4-2 lead and give that up."
Tampa Bay coach Guy Boucher called a timeout after the Lightning fell behind by two.
"Normally, he'd be pretty mad and let the guys know it," Stamkos said. "He was calm. He just said, 'Let's not allow anything else, and we know we're going to get our goals. Stay calm.'"
Stamkos scored 15 seconds into the second on a rebound goal against Emery before Sharp tied it with a power-play goal at 1:05.
"We had the momentum, we had them on their heels," Blackhawks captain Jonathan Toews said. "But we got into some bad habits and let them back into it."
The Blackhawks went 0 for 19 with the man advantage in their previous five games and failed to convert their first opportunity Friday.
Roloson made 28 saves, while Emery stopped 27 shots.
Chicago right wing Patrick Kane had three assists. He has five assists and seven points over his last three games.
The Lightning franchise record for goals in a game is four, set by Chris Kontos in Tampa Bay's first ever game, a 7-3 win over Chicago on Oct. 7, 1992. ... Blackhawks D Duncan Keith missed his second straight game due to a left hand injury. He could return early next week. ... Hall of Fame goalie Tony Esposito, who led Chicago to the 1971 and 73 Stanley Cup finals, was at the game. ... Hossa has 24 goals and 57 points in 52 games against the Lightning. ... Chicago D Nick Leddy was a minus-4.
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TAMPA, FL - NOVEMBER 4: Ryan Shannon #22 of the Tampa Bay Lightning heads to the ice for pre-game skate against...