Jason Demers also scored for the Sharks, who moved ahead of the Blackhawks into first place in the Western Conference, and snapped Chicago's four-game overall and five-game home winning streaks.
"We didn't have many chances, so we were lucky to create enough to win the game," San Jose coach Todd McLellan said. "Certainly when you look at it in the big picture, we didn't play near well-enough.
"We basically robbed one here tonight and we can chalk that one up to Nabby."
Nabokov lost a shutout bid early in the third when Kane finally scored on the Blackhawks' 34th shot to make it 2-1.
"We let Nabby do all the work," Demers said. "We can't let him do that every game. He's human, too."
Chicago's puck-possession game was aided by seven power plays. That helped boost the Blackhawks' shot total, but they failed to convert any of the chances and allowed a short-handed score by Thornton.
"Taking all the penalties slowed us down, but we battled," Nabokov said. "We needed our big guys to score, and they did."
San Jose capitalized on Chicago's mistakes to avenge 4-3 and 7-2 losses to the Blackhawks in November.
McLellan had reassembled a line of top forwards Heatley, Thornton and Patrick Marleau to help spark San Jose to a 4-2 win at Dallas on Monday. McLellan kept the trio together against Chicago and it paid off.
"They scored two big goals and set up the winner," McLellan said. "They contributed with the penalty killing. Without them we'd have no offense."
Chicago's Cristobal Huet made 11 saves, but his shutout streak ended at 143:31 when Heatley opened the scoring 3:18 into the first. Huet had 3-0 shutouts in his two previous starts.
Huet had not permitted a goal since Buffalo's Clarke MacArthur beat him late in the second period of the Sabres' 2-1 win on Dec. 11.
"We outshot them, but we couldn't create the top, top chances," Hossa said. "We tried to shoot from bad angles and chaos in front of the net, but their goalie played well.
"We have to learn from this," he added. "We have to be more poised with the puck and trust each other more."
San Jose's three-game winning streak comes on the heels of an 0-2-3 slide, its longest winless slump this season.
The Blackhawks outshot the Sharks 15-4 in the first, but trailed 1-0 after 20 minutes. Heatley put San Jose ahead early when he slipped around Chicago defenseman Brent Seabrook, took Dan Boyle's feed and beat Huet with a high shot on the short side.
In the second period, Chicago outshot San Jose 11-3, but the Sharks took a 2-0 lead on Thornton's short-handed goal at 5:10. Thornton is the NHL's leading scorer.
San Jose's Joe Pavelski picked off a pass by Chicago's Kris Versteeg at center ice, and then moved in on a 2-on-1 break with Thornton. Pavelski slipped a pass to Thornton, who lofted a rising shot over Huet's left shoulder.
Kane cut it to 2-1 at 4:10 of the third to end Nabokov's bid for a third shutout this season.
Hossa dug the puck out of the left corner, centered it and Kane knocked it in from a goal-mouth scramble.
Just 20 seconds later, Demers made it 3-1 following a Chicago coverage breakdown. Set up by Thornton's pass, Demers beat Huet on the glove side with a high shot from the slot.
Hossa slipped in a goal from a sharp angle during a 4-on-4 with 1:51 left in the third to cut it to 3-2.
The Blackhawks have converted just three of their past 36 power-play chances. ... Huet's longest personal shutout streak is 173:30, set over four games with Montreal in February 2006. His previous back-to-back shutouts were in March 2006 with the Canadiens. ... Chicago D Cam Barker missed his second game with a lower-body injury and LW Ben Eager missed his third with an upper-body injury. ... San Jose RW Brad Staubitz missed his 14th game with a facial injury. ... D Jay Leach was a healthy scratch.