It was the longest game to date in the 2013 playoffs and was the longest Game 1 of the Stanley Cup Finals in 23 years.
Oduya's goal that saved the Blackhawks came when his shot bounced off the skate of Boston defenseman Andrew Ference with 7:46 left in regulation.
With both teams obviously tiring from the more than 100 minutes of hockey, Chicago's Michal Rozsival rifled a shot from between the circles into the traffic in front of the net. Shaw got a stick on it between two defenders and it sailed past Tuukka Rask to end the game.
Boston lost for only the second time in its last 10 playoff contests, having reached the NHL's championship series by sweeping Pittsburgh.
Milan Lucic scored the game's first two goals to give Boston a 2-0 lead, but Brandon Saad lifted the Blackhawks within one 3:08 deep in the second period.
The Bruins regained a two-goal advantage when Patrice Bergeron produced a power-play goal 6:09 into the third.
Rask, who gave up only two goals in the four games against Pittsburgh, made 59 saves. Corey Crawford stopped 51 shots for the Blackhawks, who owned the best regular-season record in the NHL this year.
Chicago will host the second game of the finals Saturday.