NEW YORK — Lawsuits claiming Saudi Arabia aided al-Qaida and should be held liable for the Sept. 11, 2001, terrorist attack were revived by a U.S. appeals court in a decision that allows victims and their families another chance to seek compensation from the kingdom.

The U.S. Court of Appeals in New York Thursday said a lower-court judge “rested on an error of law” in rejecting a request to reopen the cases against the country’s government and an affiliated charity.

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The Kingdom of Saudi Arabia and the Saudi High Commission for Relief of Bosnia and Herzegovina won a ruling dismissing them from the lawsuits on the grounds that a foreign government is immune from suit.

Subsequently, the appeals court made inconsistent rulings on whether the terrorism claims made in the case fall under sovereign-immunity rule, according to yesterday’s opinion.

The lower-court judge should have granted a request to re- open the cases to allow the inconsistency to be addressed, the appeals court said.

The inconsistent rulings caused “a disparity” between two cases “where none should ever have existed,” the appeals court said.

“We conclude that the circumstances of this case are extraordinary,” warranting its re-opening the three-judge panel said in the ruling. The case will be returned to the lower-court judge for consideration as to whether it should move forward.

Michael Kellogg, a lawyer for the Kingdom of Saudi Arabia, didn’t immediately return a call yesterday seeking comment on the ruling.

Jerry S. Goldman, a lawyer for the plaintiffs, said in a statement that the decision is “soundly grounded and restores this case to the proper procedural posture.”

“This is a big step forward in the process of obtaining fair justice for the victims of this tragedy,” he said.

Goldman’s clients include family members of John O’Neill, a former federal counter-terrorism agent who had led investigations of Osama bin Laden and was working as the chief of security for New York’s World Trade Center when the attacks occurred.

Saudi Arabia is the world’s biggest source of funds for militant Islamic groups such as the Taliban in Afghanistan and Pakistan’s Lashkar-e-Taiba, and the Saudi government is reluctant to stem the flow of money, then-Secretary of State Hillary Clinton said in a 2009 cable obtained by Wikileaks.

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Donors in Saudi Arabia constitute the most “significant source of funding to Sunni terrorist groups worldwide,” according to the cable.

Drew C. Bailey, a spokesman for the state department, declined to comment on the cable.