The House voted 419-3 to pass the bill that seeks to punish Russia for allegedly meddling in the 2016 presidential elections.
Other than Russia, the legislation also seeks to make Iran pay for its "continued support of terrorism", and includes provisions to curb North Korea's nuclear programme.
The bill has in recent weeks attracted controversy.
However, Press Secretary Sarah Sanders told reporters: "The White House, the president, and the entire administration strongly support sanctions against Russia, Iran, and North Korea."
The bill now heads to the Senate before it can be sent to the White House for Trump to sign it into a law.
If enacted, it could put strain on Trump's ability to improve ties with Russia, which he has vocally pursued, but has been restrained by the allegations that his associates had contacts with Russian officials during the election campaign.
Congressman Ed Royce, chairman of the House Foreign Affairs Committee, and Senate minority leader Chuck Schumer have urged the Senate to pass the legislation without delay.
"Yesterday, the House of Representatives passed, nearly unanimously, a sanctions bill that was a product of bicameral, bipartisan negotiations and includes strong sanctions against Russia, Iran, North Korea. The Senate must act quickly on the legislation from the House," Schumer said.
Senator and chairman of the Foreign Relations Committee Bob Corker has indicated he plans to strip out a section of this package that relates to North Korea, he said, calling it "yet another delay" by Republicans to prevent this bill from landing on the Trump's desk before "we leave for the recess."
"I expect this bill will pass the Senate in short order. At that point, it is incumbent upon President Trump to immediately sign this legislation into law, or risk endorsing Moscow's interference in future elections," he said.
"We must also make clear to Iran and North Korea that they will be held accountable for their actions," Warner said.
Later, Corker announced a path forward on the bill.
"The Senate will move to approve the Iran and Russia sanctions it originally passed six weeks ago, as well as the North Korea sanctions developed by the House," he added.
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