The House of Representatives passed the measure unanimously two days after the Senate approved it.
Hamid Aboutalebi was a part of the Muslim student group that seized the US embassy in Tehran in 1979.
The White House has told Iran Mr Aboutalebi was "not viable" but has not taken a position on the bill.
Fifty-two Americans were held for 444 days at the height of Iran's Islamic revolution, which saw pro-American Shah Mohammad Reza Pahlavi sent into exile and Ayatollah Ruhollah Khomeini take power.
Mr Aboutalebi, who previously served as Iran's ambassador to Belgium, the European Union, Italy and Australia, told Iranian media his participation in the hostage crisis began only after the initial seizure of the embassy, and primarily involved translation.
On Thursday, White House spokesman Jay Carney said, "We've made clear and have communicated to the Iranians that the selection they've put forward is not viable."
But he declined to say whether President Barack Obama would sign the bill into law.
The Iranian government, meanwhile, has called the US rejection of Mr Aboutalebi "not acceptable".
The bill passed on Thursday in the House denies entry into the US to individuals engaged in espionage or terrorism or who pose a threat to national security.
Senator Ted Cruz, the Texas Republican who introduced the legislation in the US Senate, urged Mr Obama to sign the bill.
"We, as a country, can send an unequivocal message to rogue nations like Iran that the United States will not tolerate this kind of provocative and hostile behaviour," Mr Cruz said.
The bill's sponsor in the House, Republican Doug Lamborn, said, "Terrorists, from Iran or elsewhere, should not be allowed to walk the streets of Manhattan with diplomatic immunity."
As the host country of the United Nations, the US has previously but rarely denied entry to an envoy or head of state. Those included a previous Iranian diplomat and Sudanese President Omar al-Bashir.
In those cases the applications were withdrawn after the US signalled opposition, or the state department simply declined to process the visas.
Those options are available in Mr Aboutalebi's case.
On Monday, Iranian foreign ministry spokeswoman Marzieh Afkham told state-run media he had previously received a US visa as part of a visit to the UN in the 1990s.
In an interview with an Iranian news site, Mr Aboutalebi said he was not part of the group that took over the US embassy and was only later asked to translate for the students.
Others involved in the hostage-taking have corroborated that account.

Over six years, the Tackling FGM Initiative has given Â£2.8m of funding to community-based prevention schemes.
It said in a report that the funding had helped spark a necessary debate about FGM.
But it added that maintaining momentum would be difficult without more money.
The initiative was set up by five charitable trusts, including Comic Relief, to discourage the cutting of girls' sexual organs.
As well as funding the work of dozens of community groups, the money also paid for the training of more than 6,000 teachers and healthcare staff in how to spot and respond to the illegal act of female genital mutilation.
Under the scheme, people were encouraged to work within their own communities. For example, a Somali elder in Bolton recruited other men in his community to emphasise the need to protect their daughters.
But the Tackling FGM Initiative's report said there was still a lack of clarity over who should be funding future FGM prevention work.
It said local authorities, health and wellbeing boards, and public health or clinical commissioning groups all had a part to play.
The report says it is difficult to measure the success of preventing FGM because of the lack of data about an illegal procedure, but, using surveys, it found attitudes to FGM had changed.
It concludes that awareness of FGM has increased, particularly among women, and there has been a strong focus on educating young people from a wide range of communities.
But the report warns that funding is still the major barrier to ending mutilation and the authors say, in the current financial climate, it will be difficult to maintain progress.
It said: "There is good evidence that the community-based approach has worked with a range of audiences within communities affected by FGM and has started to create a critical mass of people who are opposed to the practice."
The US Congress has sent a bill to the president that would bar Iran's pick for ambassador to the UN from entering the country.
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Community groups involved in an initiative to tackle female genital mutilation in England and Wales have warned that future progress could be threatened by a lack of local funding.