Just two minutes into the second period, St. Louis had matched the club mark with four goals on the game. The Lightning had the lead, but it wouldn't last.
Pavelski beat goalie Ben Bishop twice in a span of 21 seconds midway through the period, and then added the winner a minute before the intermission.
St. Louis struck three times against goalie Antti Niemi in a span of 7:34 in the first period for his eighth NHL hat trick. His fourth goal early in the second equaled the team mark set by Chris Kontos against Hockey Hall of Fame goalie Ed Belfour and the Chicago Blackhawks in Tampa Bay's inaugural game on Oct. 7, 1992.
St. Louis scored twice on the power play against a penalty-killing unit that hadn't allowed a goal in six consecutive games. His four goals were one more than San Jose had allowed in its previous four games combined.
''I did appreciate them, until we were down and the game was over,'' St. Louis said. ''It's tough to really be happy about it (when) at the end of the day you lose the game.''
Tampa Bay sustained its fifth straight one-goal loss at home.
''Spotting him with four goals, it's lucky we won the game,'' Sharks captain Joe Thornton said. ''He's been a great player for a lot of years. He knows what to do when he gets the puck.''
Thornton also reached a milestone, joining Hall of Famer Bobby Hull in 48th place on the NHL career list with 1,170 points with his assist on Pavelski's winner.
''Right place right time,'' Thornton said. ''He just has a knack for finding the back of the net. For us to come back down 4-2, it was big for us.''
Matt Nieto and Patrick Marleau put San Jose ahead 2-1 with goals three minutes apart midway through the first period. St. Louis scored his first goal in between, then struck twice more before the intermission.
His record-tying score 2:09 into the second period gave him eight goals and 11 points in the NHL's longest current point streak at seven games.
''I thought we put ourselves in a good spot, up 4-2 against a pretty good San Jose team,'' St. Louis said. ''We didn't keep pushing, we took penalties, and next thing you know, we are down a goal, so it's deflating. We just didn't get it done.''
NOTES: San Jose has won eight of 10 in the series. . Bishop earned his second assist of the season from the Tampa Bay net. ... Lightning D Victor Hedman was helped off the ice after sustaining a lower-body injury early in the third period. . Tampa Bay fell to 7-1-1 at home against Western Conference foes. . St. Louis entered the game one point behind Hall of Famer Larry Robinson on the NHL career list. He also vaulted past Kirk Muller into 86th place with 961. . The Sharks are 22-2-5 when they score first.