The official said the 31-year-old protester died on Friday, a few hours after he was shot at the border where Palestinian and Lebanese youth had gathered as part of a rally against the Israeli military campaign in Gaza.
A small group of them later crossed the border into Israel, triggering the shooting.
The Israeli military says troops fired warning shots toward the group after they sabotaged the fence and crossed over briefly.
There was no immediate comment from Lebanese officials or the military on the incident.
The police official spoke on condition of anonymity in line with regulations.