Henderson put Charlotte in front for good by hitting a jumper with just 37 seconds remaining and then scored on a dunk with 29 seconds to play.
Two free throws by Kemba Walker with 21 seconds left finished the scoring and brought Charlotte its 18th victory of the season -- fewest in the NBA.
The Bobcats ended a three-game losing streak and handed the 76ers their first loss in four outings. Philadelphia had been trying for its first four-game winning streak of the season.
Henderson finished with 24 points while Walker contributed 17 points and 10 assists.
Philadelphia was led by Damien Wilkins with 20 points. The 76ers' Jrue Holiday put up 24 shots from the field and made just two of them.