The move prompted howls of protest from Democrats and some Russia hawks, who accused the administration of going soft on Russia in the midst of the special counsel’s investigation into whether the Trump campaign coordinated with Russia’s election meddling. Mr. Deripaska has emerged as a bit character in the story lines around the investigation as a result of his payments to Paul Manafort, Mr. Trump’s former campaign chairman, who has been convicted and pleaded guilty to charges brought by the special counsel, Robert S. Mueller III.

Democrats urged the Treasury Department to postpone the final decision on sanctions relief for Mr. Deripaska’s companies until after Mr. Mueller’s investigation concludes.

And the party’s congressional leaders began pushing resolutions in the Senate and the House this week to override the sanctions relief. The resolutions were introduced under a provision in a 2017 law that allows Congress to override certain sanctions-related decisions by the Treasury Department. Under the provision, both the House and the Senate would need to pass resolutions by a Thursday deadline to block the administration’s move to lift the sanctions. Failing congressional action, the sanctions would automatically be lifted.

The Democrats’ efforts at first seemed to stand little chance, given efforts to stifle them by the administration and the Republicans who control the Senate.

Mr. Mnuchin visited Republican senators on Tuesday before the vote to implore them to support the sanctions relief. The appearance followed a briefing to the House last week in which he argued that the deal his agency negotiated with Mr. Deripaska’s companies would punish the oligarch by separating him from the companies, without having economic ripple effects. The briefing left Democratic leaders unconvinced.

And Tuesday’s Senate vote indicated that Mr. Mnuchin’s assurances did not assuage doubts among the party’s Russia skeptics that the deal with Mr. Deripaska’s companies sufficiently decreased his control.

“I don’t like the way it’s structured,” said Senator Marco Rubio, Republican of Florida, who voted with Democrats and has been among the party’s leading critics of Russia.