The measure, which passed on a vote of 98 to two, seeks to make Tehran pay a price for its "continued support of terrorism."
It also aims to punish Russia's Vladimir Putin for interfering in last year's US election, and to make it tougher for the White House to roll back sanctions.
US intelligence chiefs have concluded that Russia orchestrated a campaign to undermine the American election process that included espionage and cyber-attacks, as a means to tilt the vote in Trump's favor.
"Any idea of the president's that he can lift sanctions on his own for whatever reason are dashed by this legislation."
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The bill as originally introduced was exclusively about slapping new sanctions on Iran. But lawmakers attached a bipartisan amendment on Russia to it early this week.
The measure would require a green light from Congress in the event sanctions on Russia are relaxed, suspended or terminated.
It would codify in law the sanctions imposed by executive decree by Trump's predecessor Barack Obama, especially against the Russian energy industry.
And it would impose new sanctions on "corrupt Russian actors," those implicated in serious human rights abuses or who supply weapons to Bashar al-Assad's regime in Syria, and people who conduct "malicious cyber activity" on behalf of the Russian state.
Corker, too, sounded pleased that the bill effectively ties a president's hands when it comes to unwinding certain sanctions on Russia.
"Today the United States Senate is asserting its responsibilities" regarding foreign policy, he added.