It was Toews' second goal in three games. Of Chicago's last 10 goals, Chara was on the ice for nine.
Boston right wing Jaromir Jagr was shaken up in the first period. He returned for the second but left the bench, and Tyler Seguin replaced him on the second line with left wing Brad Marchand and center Bergeron.
The play that led to Kelly's goal began after a faceoff that rookie defenseman Torey Krug rushed in to tip toward a teammate. The puck went to Daniel Paille, standing about 40 feet on the left. He passed to Seguin, who caught the puck with his right glove in the slot and dropped it.
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 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 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.