Seguin then passed to Kelly, who scored his second goal of the playoffs 7:19 into the game.
It came just seven seconds after a whistle stopped a scrum in front of the net that followed an extended period of pressure by the Bruins.
Just two minutes after the goal, Chicago had one of its best chances of the period when Michal Frolik skated in with the puck behind the defense and fired a 15-foot drive from the left, but Rask made the save.
Boston had another solid chance at 12:24 when Milan Lucic took a 15-foot shot from the slot that Crawford stopped.
After having no power plays in Game 5, the Bruins had four failed advantages in the first two periods.
With 4:01 left in the first, Shaw was struck in the face by a puck when it deflected off the shaft of his stick after Boston's Shawn Thornton shot it. He lay on the ice before getting up and skating off slowly.
Toews was on Chicago's first shift of the game. Bergeron had left Game 5 with an undisclosed injury after playing just 49 seconds in the second period.
Five of the last nine Cup finals have gone seven games, including in 2011 when the Bruins overcame a 3-2 series deficit and won their first championship since 1972 by winning Game 6 in Boston and Game 7 in Vancouver.
In 2010, Chicago won its first NHL title since 1961 on Kane's overtime goal. As they did this year, the Blackhawks won Game 5 to take a 3-2 series lead.
This year's finals have been extremely tight, with three of the first five games going to overtime. Chicago won the opener in three overtimes, then Boston won 2-1 in one extra period and 2-0. The Blackhawks regained home-ice advantage with a 6-5 overtime win in a wild Game 4 in Boston before returning home for Saturday night's win.
Teams that have won Game 5 after splitting the first four have won the Cup 15 of 22 times since the best-of-seven format began in 1939. But the loser of Game 5 the past six times has won four championships, including the Bruins against the Canucks.
Last season, the Los Angeles Kings beat the New Jersey Devils in six games. This season, the Blackhawks beat the Kings in five games to reach the Cup finals, clinching the series on Kane's goal in overtime.