Residents are subject to border closures and Israel recently moved to cut fuel and electricity supplies in response to the firing of homemade rockets into southern Israel.
Police displayed the drugs at a news conference then made a bonfire and set fire to them.
Most of the drugs were thought to have been smuggled into Gaza from Egypt although some cannabis was grown locally.
Ehab al-Ghsain, a spokesman at the Hamas-run interior ministry, said dealers could get a life sentence or even the death penalty under Palestinian law.
After its fighters routed Fatah forces in June, Hamas said it would crack down on crime among the population of almost 1.5 million.
Abbas has mounted his own security drive in the larger West Bank, where he holds sway, to impose order as the Palestinians start formal statehood talks with Israel.
Alcohol is virtually impossible to get in Gaza, but some young men use drugs, even though it is frowned upon in the conservative society.