Two armed boats approached the Seabourn Spirit about 100 miles off the coast of Somalia and fired as the boats' occupants attempted to get onboard, said Bruce Good, a spokesman for Miami-based Seabourn Cruise Line, a subsidiary of Carnival Corp.
The ship outran them and changed its course.
"Our suspicion at this time is that the motive was theft," Good said, adding that the crew had been trained for "various scenarios, including people trying to get on the ship that you don't want on the ship."
CBS News correspondent Randall Pinkston reports that while the ship's captain took evasive maneuvers, the passengers were ushered below deck to the ship's dining room.
"He was, to some extent, steering the ship to create waves to overturn the boats and at one point he nearly rammed one of the boats," said passenger Norman Fisher.
The attackers never got close enough to board the Spirit, but one member of the 161-person crew was injured by shrapnel, said Debrah Natansohn, president of the cruise line.
Press Association, the British news agency, said passengers awoke to the sound of gunfire as two 25-foot inflatable boats approached the liner.
Edith Laird of Seattle, who was traveling on the ship with her daughter and a friend, told British Broadcasting Corp. TV in an e-mail that her daughter saw the pirates out of their window.
"There were at least three rocket-propelled grenades that hit the ship, one in a state room," Laird wrote. "We had no idea that this ship could move as fast as it did and (the captain) did his best to run down the pirates."
None of the vessel's 151 passengers, mostly Americans with some Australians and Europeans, were injured, Good said.
The Spirit had been bound for Mombasa, Kenya, at the end of a 16-day voyage from Alexandria, Egypt. It was expected to reach the Seychelles in the Indian Ocean on Monday, and then continue on its previous schedule to Singapore, company officials said.
The 10,000-ton cruise ship, registered in the Bahamas, sustained minor damage, Good said. "They took some fire, but it's safe to sail," he said.
Piracy along the Somalia coast is common — several ships a month are attacked or hijacked, with valuables stolen and crews held for ransom.
Somalia's 1,880-mile coastline is Africa's longest, and the country has had no effective central government since opposition leaders ousted dictator Mohamed Siad Barre in 1991. They then turned on each other, transforming this nation of 7 million into a patchwork of battling fiefdoms ruled by heavily armed militias.