Israel circulated a picture of what it said was the ax lying on the ground near the border.
A second Palestinian who was with the man fled the scene after the shooting, Reuters quoted an Israeli military spokesman as saying.
There was no immediate comment from Palestinian officials on the incident, which comes at a time of renewed violence along the Israel-Gaza frontier.
Since March 30 Israeli troops have killed at least 120 Palestinians during mass demonstrations along the Gaza border, drawing international condemnation.
The Palestinians say most of the dead and the thousands wounded were unarmed civilians against whom Israel was using excessive force.
More than 2 million Palestinians are packed into Gaza, a narrow coastal enclave where unemployment is high and poverty widespread.