JERUSALEM—White House and State Department officials who were leading U.S. efforts to rein in Israel's military campaign in the Gaza Strip were caught off guard last month when they learned that the Israeli military had been quietly securing supplies of ammunition from the Pentagon without their approval.

Since then the Obama administration has tightened its control on arms transfers to Israel. But Israeli and U.S. officials say that the adroit bureaucratic maneuvering made it plain how little influence the White House and...