The teams were tied at 89-89 as the game entered its final half minute, but Henderson gave Charlotte the lead for good with a dunk.
Two free throws each by Josh McRoberts and Michael Kidd-Gilchrist increased the Bobcats' advantage to 95-89 and gave them their second straight win.
The Clippers had won six of their previous seven and fell to 2-2 on their seven-game road trip.
Al Jefferson had 24 points and 10 rebounds for Charlotte, which allowed Los Angeles just nine free throws.
2014: The Year in Music [PHOTOS]