The sanctions bill comes as politicians investigate possible meddling by Russia in the 2016 presidential election and potential collusion by Republican Trump's campaign.
New sanctions against Iran and its Islamic Revolutionary Guard Corps, which stands accused of supporting "terrorism"; and North Korea, for its missile tests, are also included in the bill backed by house members.
Russia and Iran were quick to respond to the vote.
Russian Deputy Foreign Minister Sergei Ryabkov told the Interfax news agency the move left no room to improve ties between Moscow and Washington in the near future and took the relationship into uncharted waters.
"This is already having an extremely negative impact on the process of normalising our relations," Ryabkov told Interfax.
A prominent member of the upper house of Russia's parliament, Konstantin Kosachyov, said that Moscow should prepare a "painful" response to new US sanctions.
"Judging by the unanimous vote in the US House of Representatives on the sanctions package against Russia, Iran and North Korea, there will be no breakthrough [in U.S.-Russian relations] ... In fact, further degradation of bilateral cooperation is becoming inevitable," Kosachyov said on his Facebook page.
According to Iranian state media, Deputy Foreign Minister Abbas Araqchi decried the bill as "a completely clear hostile act" that "will be met with a definitive response", though he did not specify what the actions would be.
'Threatening vital US interests'
The Republican-controlled Senate passed an earlier version of the bill with near-unanimous support.
The House of Representatives added the North Korea measures after becoming frustrated with the Senate's failure to advance a bill it passed in May.
Representative Ed Royce, the Republican chairman of the House Foreign Affairs Committee, said the three countries "are threatening vital US interests and destabilising their neighbours. It is well past time that we forcefully respond".
It was unclear how quickly the bill would make its way to the White House for Trump to sign into law or veto.
The bill still must be passed by the Senate, which is mired in debate over efforts to overhaul the US healthcare system as politicians try to clear the decks to leave Washington, DC, for their summer recess.
Bill 'likely to pass'
Al Jazeera's Heidi Zhou-Castro reporting from Washington, DC, said the bill is likely to pass in the Senate and be on the president's desk for final approval by the end of the month.
"This was a rare show of bipartisan solidarity in a near unanimous vote and reflects the widespread concern about Trump's perceived overfriendly relationship with Moscow," she said.
"If Trump signs the bill, giving it approval, he will acknowledge that Russia did meddle in 2016 US presidential elections.
"But if he does not sign the bill, he faces a political firestorm given that his campaign, his close family members and closest advisers are under investigations in Capitol Hill for their alleged roles of colluding with the Russian government in getting Trump to the White House."
The intense focus on Russia, involving several congressional probes and a separate investigation by a Justice Department-appointed special counsel, Robert Mueller, has overshadowed Trump's agenda.
The scrutiny has angered and frustrated the president, who calls the investigations a politically motivated witch-hunt fuelled by Democrats who cannot accept his upset win in last November's election against Democratic candidate Hillary Clinton, a former secretary of state.
On Tuesday, Trump's son-in-law, Jared Kushner, spent three hours with the House of Representatives intelligence panel, his second straight day on Capitol Hill answering questions about his contacts with Russians during the campaign.
Kushner had a "very productive session" with the House Intelligence Committee, Democratic Representative Adam Schiff said after the meeting.
Kushner, who is now a top aide in Trump's White House, told reporters on Monday he had no part in any Kremlin plot.
US House Republicans on Tuesday rejected a legislative effort by Democrats to obtain Treasury Department documents that could show any ties between the finances of Trump, his inner circle and the Russian government.
Source: Al Jazeera and news agencies