“One of my goals as the leading Republican on the Senate Foreign Relations Committee is to recapture much of the Senate prerogatives on foreign policy,” said Senator Bob Corker of Tennessee, the committee chairman. He added that the panel had “dissipated for a long time into a debating society.”

Congress has struggled with what to do about Russian interference in the 2016 presidential campaign and aggression in Eastern Europe since Mr. Trump was elected. Some early efforts at new sanctions on Russia were considered too tough by some Republicans. After the election, Mr. Corker waited for Mr. Tillerson, at Mr. Tillerson’s request, to attempt a relationship with his Russian counterpart, Sergey V. Lavrov, the Russian foreign minister.

But Mr. Corker decided the effort was fruitless and called Mr. Tillerson from a secure location in the Capitol to let him know that he was proceeding with a strong Russia sanctions bill. That was not an option Mr. Tillerson preferred, Mr. Corker said in an interview.

“What went into passing that bill was incredibly intense,” Mr. Corker said. “Our staff worked around the clock to get us where we are, in a place where the election passions had dissolved into significant policy discussions.”

Senator Benjamin L. Cardin of Maryland, the highest-ranking Democrat on the committee, agreed. “This is a very important moment for this Congress,” he said. “What we recognized as we were working on this is we were not really talking about President Trump, we were talking about American values. If you don’t say, ‘This is to counter Trump,’ even if you are in fact countering Trump, that’s how you get Republican partners.”

The bill — which still must clear the House after the 97-to-2 Senate vote — empowered Congress to block the president from reducing existing sanctions and added new ones in response to Russia’s support for President Bashar al-Assad of Syria and Kremlin interference in last year’s election.