The Senate passed the measure by voice vote in May, but the administration has argued it would complicate diplomatic relations with a key ally in the region and warned against moving it forward.
The vote to send the bill to the President comes two days before the 15th anniversary of the attacks that killed nearly 3,000 Americans.
House Speaker Paul Ryan was pressed about the pushback from some in the diplomatic and legal community about the precedent it would set, but said, "this bill passed overwhelmingly in the US Senate
so I think that those concerns have been taken under consideration and members are acting accordingly and that's why this bill will pass."
The White House had no comment on the House's action Friday.
After the Senate bill passed in May, White House spokesman Josh Earnest said "it's difficult to imagine the President signing this legislation."
"I know that the advocates of this legislation have suggested that they have taken into account our concerns by more narrowly tailoring the legislation," Earnest said. "But, unfortunately, their efforts were not sufficient to prevent the longer-term, unintended consequences that we are concerned about. This legislation would change longstanding international law regarding sovereign immunity. And the President of the United States continues to harbor serious concerns that this legislation would make the United States vulnerable in other court systems around the world."
House Judiciary Chairman Bob Goodlatte, R-Virginia, refuted that the bill interfered with the sovereign immunity of other countries, and said international acts of terrorism deserve to be exceptions in terms of legal liability.
"We can no longer allow those who injure and kill Americans to hide behind legal loopholes, denying justice to the victims of terrorism," Goodlatte said.
"There are always diplomatic considerations that get in the way of justice, but if a court proves the Saudis were complicit in 9/11, they should be held accountable," said Sen. Chuck Schumer, D-New York, a sponsor of the Senate bill. "If they've done nothing wrong, they have nothing to worry about."
Saudi Arabia has denied any role in the attacks and has never been formally implicated, but 15 of the 19 hijackers that carried them out were of Saudi descent. The Saudi foreign minister, Adel al-Jubeir, has warned lawmakers
that if the bill became law, the country would sell $750 billion in US assets, including treasury securities.
Sen. John Cornyn, R-Texas, a co-sponsor of the Senate bill, said in a statement that "today's vote sends an unmistakable message that we should combat terrorism with every tool we have, and that the families of those lost in attacks like that on September 11th should have every means at their disposal to seek justice."
Pointing to the concerns raised by the White House, Michigan Rep. John Conyers, the top Democrat on the Judiciary panel, said Friday, "I remain hopeful that we can continue to work with the Administration so we can resolve these issues so the legislation can be signed into law by the president."
House Minority Leader Nancy Pelosi previously acknowledged that after the Senate easily passed the bill the administration reached out to try to make some changes but said it was "a little late."
Recently unclassified documents detail contact and support between some of the hijackers and individuals who may have been connected to the Saudi government. The "28 pages" that were long secret also note suspicions about ties between the Saudi royal family and al Qaeda, though the documents also say the speculations have not been verified.