Michal Rozsival started the winning sequence with a shot from the right point into traffic. Bolland tipped it before Shaw got it by Rask for his fifth goal of the playoffs.
And just like that, the longest Finals game in five years was over.
"We knew it wasn't going to be pretty," Shaw said. "It was a great shot, great shift. It was unbelievable. All the guys deserved this. It was a great battle for us."
Milan Lucic had two goals and an assist for the Bruins, who had won five straight and nine of 10. Patrice Bergeron scored a power-play goal and David Krejci finished with two assists.
Rask made an astounding 59 saves in the longest Finals game since Pittsburgh beat Detroit 4-3 when Petr Sykora scored at 9:57 of third overtime on June 2, 2008.
Game 2 is Saturday night at the United Center.
"Get some rest, because we basically just played two games in one night here," Blackhawks coach Joel Quenneville said. "But based on how it went tonight, we'll have to be ready."
The Blackhawks won for the eighth time in nine games and improved to 10-1 at home in the playoffs. Bolland and Johnny Oduya scored in the third period for Chicago, which never would have made it to the third overtime if not for an impressive performance by goaltender Corey Crawford. Brandon Saad had his first goal of the playoffs.
Crawford gave the Blackhawks a chance by standing his ground when the Bruins had repeated opportunities in the extra sessions.
Shawn Thornton and Daniel Paille skated in for a 2-on-1 in the first OT, and Crawford turned away Thornton on the doorstep. He denied Rich Peverley and Tyler Seguin in rapid succession, and helped Chicago kill off two power plays when it was whistled twice for too many men on the ice.
Nathan Horton hit the post in the first extra session, and Zdeno Chara's slap shot deflected off Jaromir Jagr and then the inside of the right post at the very end of the second overtime.
Crawford had 22 of his 51 saves in the first two overtimes, and Rask was forced to make 18 stops. The action was so fast and furious that it took a toll on the players with Horton skating off during the power play with an injury likely a serious one to leave the ice during that pivotal moment.
"Not disappointed in our effort," Bruins coach Claude Julien said. "There's certain things you're going to want to fix for next game. But as far as the game is concerned, it was a hard-fought game."
All games at 6 p.m.
Wednesday • Chicago 4, Boston 3 (3OT)
Blackhawks lead 1-0
Saturday • Boston at Chicago, NBCSN
Monday • Chicago at Boston, NBCSN
June 19 • Chicago at Boston, Ch. 5
x-June 22 • Boston at Chicago, Ch. 5
x-June 24 • Chicago at Boston, Ch. 5
x-June 26 • Boston at Chicago, Ch. 5
Blackhawks 4, Bruins 3
Boston 1 1 1 0 0 0 3
Chicago 0 1 2 0 0 1 4
First Period1, Boston, Lucic 4 (Horton, Krejci), 13:11. PenaltiesNone.
Second Period2, Boston, Lucic 5 (Krejci), :51. 3, Chicago, Saad 1 (Hossa), 3:08. PenaltiesHorton, Bos (interference), 7:37; Boston bench, served by Thornton (too many men), 8:20; Chara, Bos (high-sticking), 12:53.
Third Period4, Boston, Bergeron 6 (Seguin, Lucic), 6:09 (pp). 5, Chicago, Bolland 1 (Shaw), 8:00. 6, Chicago, Oduya 3 (Kruger, Frolik), 12:14. PenaltiesFrolik, Chi (tripping), 5:51.
First OvertimeNone. PenaltiesChicago bench, served by Sharp (too many men), 12:08.
Second OvertimeNone. PenaltiesChicago bench, served by Shaw (too many men), 19:07.
Third Overtime7, Chicago, Shaw 5 (Bolland, Rozsival), 12:08. PenaltiesNone.
Shots on GoalBoston 11-6-8-12-10-754. Chicago 8-16-15-8-10-663.
Power-play opportunitiesBoston 1 of 3; Chicago 0 of 3.
GoaliesBoston, Rask 12-5-0 (63 shots-59 saves). Chicago, Crawford 13-5-0 (54-51).
A22,110 (19,717). T4:38. RefereesChris Rooney, Brad Watson. LinesmenShane Heyer, Brian Murphy.