CAIRO, Egypt—It seems inconceivable: Somali pirates in speedboats foil warships from the world’s most powerful navies to prey on shipping lanes crucial to the oil supply.

How do they do it? Basically, it’s a big ocean and no one wants to be top cop.

NATO and the U.S. Navy say they can’t be everywhere, and American officials are urging ships to hire private security. Warships patrolling off Somalia have succeeded in stopping some pirate attacks. But military assaults to wrest back a ship are highly risky and, to this point, uncommon.

Yet when pirates took their biggest prize to date over the weekend—a Saudi supertanker carrying $100 million worth of crude oil—it raised the stakes dramatically. The pirates struck hundreds of miles off the coast of East Africa, far out to sea where ships were presumed to be safe.

Governments, navies, oil companies and ship owners are scrambling for solutions, and finding few options are ideal. At least one private security company said it has been flooded with requests from shipping companies for protection, including from Saudis.

Shippers and insurance companies that once minimized piracy’s risks—at worse, pay a ransom, get your ship, crew and cargo back unharmed— have now awakened to the potential economic impact. The turmoil could force cargo ships and tankers to take longer routes around Africa, drive up costs and, though it hasn’t happened yet, even raise oil prices.

“For a long time I thought this piracy thing was complete nonsense ... isolated incidents being blown up,” Giles Merritt, director of the Security & Defense Agenda, a Brussels, Belgium-based think tank on security issues.

Not anymore, he said.

“It’s unbelievable to me that we can run AWACS aircraft that can tell you anything that is moving,” Merritt said. “But apparently we cannot spot small boats full of chaps with machine guns.” Somali pirates, given free rein in a country with no stable government for two decades, have attacked more than 90 vessels this year and successfully seized 36, everything from ships carrying palm oil and chemicals to luxury yachts. They have raked in millions of dollars in ransoms and are negotiating for more than 14 vessels currently anchored at their strongholds along the coast.

Pressure is on for more warships to patrol. Saudi Arabia’s foreign minister, Prince Saud al-Faisal, said Tuesday his country was ready to join an international effort. The kingdom’s navy has 18,000-20,000 personnel, but has never taken part in any high-seas fighting.

Officials from Djibouti, Egypt, Jordan, Saudi Arabia and Sudan are to meet Thursday in Cairo to hammer out an anti-piracy strategy. But NATO, the U.S. Navy and other militaries say it isn’t as easy as sighting a pirate speedboat and intercepting it.

They say their radar does spot pirates on the prowl and they alert crews of threatened ships. But the vast stretch of the Indian Ocean and Gulf of Aden is simply too large. Warships cannot escort every ship and cannot always get to an attack scene in time.

Their focus has been the Gulf of Aden, between Somalia and Yemen, where 20,000 merchant ships a year pass on the way in and out of the Suez Canal, the quickest route from Asia to Europe and the Americas.