There was no immediate claim of responsibility for the attack, which took place before dawn when no children were present.
Islamic extremists have accused the main U.N. aid agency here of corrupting Gaza's youth with its summer program of games, sports and human rights lessons for 250,000 children.
The U.N.'s main competitor in running summer camps is the ruling Islamic militant Hamas, which says it reaches some 100,000 children. The Hamas camps teach Islam and military-style marching, along with swimming and horseback riding.
In a statement on Monday's attack, the U.N. said that about two dozen armed and masked men targeted a seaside day camp in central Gaza, one of dozens of U.N. sites across the Palestinian territory.
The assailants arrived at about 2:30 a.m., tied up four unarmed guards, then tried to set fire to two tents and a perimeter fence made from tarp-like material. Wielding knives, they slashed a plastic swimming pool, an inflatable slide and toys.
Hamas condemned the attack in a statement by the Gaza police and said it would be investigated.
John Ging, the local U.N. chief, called it a "cowardly and despicable" attack and said the site would be rebuilt quickly. The about 2,000 children who attend the targeted facility joined another camp and will remain there until theirs is rebuilt, U.N. officials said.
Ging said the attack was the latest sign of growing levels of extremism in Gaza. He said it provides further evidence "of the urgency to change the circumstances on the ground that are generating such extremism."
Ging has repeatedly called for lifting of the 3-year-old border blockade of Gaza by Israel and Egypt, arguing the sanctions have mostly hurt ordinary Gazans, not Hamas. The blockade was imposed after the violent Hamas takeover of Gaza in 2007.
Last week, Israel said it would significantly ease the blockade, responding to international pressure following its deadly raid on a blockade-busting flotilla last month.
U.N. camps have been targeted by suspected Islamic extremists since their launch in 2007.
Last month, assailants vandalized one of the main camp sites, and last year a Hamas legislator waged a public campaign against the U.N. summer program. Gaza police said they have arrested five suspects in the previous camp attack, but would not give details about their affiliation.
In recent months, Hamas has become increasingly assertive in trying to impose its strict version of Islam on everyday life in Gaza.
However, the Islamists have also avoided open confrontation with the U.N. Relief and Works Agency, which provides services for two-thirds of Gaza's 1.5 million people.
The U.N. avoids contact with Hamas, which is internationally shunned as a terror group.