Bryant's free throw with 37 seconds left gave Los Angeles the lead and the Lakers held on to win for the first time in four games since Bryant's return from Achilles tendon surgery.
Los Angeles found itself trailing by six points with 2:50 remaining but then went on a 9-0 run to end the game.
Nick Young connected on a 3-pointer, cutting the deficit to three, and Bryant followed with a layup to make it 85-84 with 1:22 to go.
After a timeout, Henderson missed a short jumper, and the Lakers' Pau Gasol made two free throws with 6.2 seconds to go.
Charlotte's Ben Gordon couldn't get a desperation 3-pointer to fall at the buzzer.
Kemba Walker provided 24 points and eight assists for the Bobcats.
Notable deaths of 2014 [PHOTOS]