But Republicans were quickly persuaded.

“That was when it really, really hit me that things were changing,” said Rob Jesmer, a top Republican strategist who was running the National Republican Senatorial Committee. Mr. Brown went on in early January to beat Martha M. Coakley, the Democratic state attorney general who backed the Obama agenda and made several costly political missteps.

This year it is Roy S. Moore, the Republican Senate candidate in deeply conservative Alabama, who is in trouble in a race that should be a gimme in such a state. Multiple accusations against Mr. Moore of sexual abuse and pursuit of teenage girls while an adult have opened the door to the previously unthinkable idea that Mr. Jones, a former federal prosecutor, could claim the seat on Dec. 12. That outcome would cut the Republican majority to 51, give Democrats a real chance to compete for Senate control next year and make it tougher to pass the tax bill this year.

While Senate Republicans hope to pass their version of the tax bill as early as next week, it would still have to be reconciled with the House plan. If Mr. Jones wins, Republicans could then afford to lose only one vote if he were seated before any final approval. Multiple Republicans remain uncommitted on the tax plan.

It is quite reminiscent of what happened when Mr. Brown’s election in January 2010 upended Democratic plans for the health care bill, which had passed the Senate on Christmas Eve. The victory by Mr. Brown, who campaigned against the health care law, deprived Democrats of the crucial 60th vote to break a filibuster. As a result, the House had to ultimately accept the Senate version of the law, preventing Democrats from making changes that would have avoided some of the problems that later dogged the legislation.

Democrats say the tax bill being pushed by Republicans desperate for their first big legislative accomplishment could blow up on them just as the health care bill cost Democrats.

“They think this is their salvation, but it is most likely their Alamo,” said Senator Chris Van Hollen, Democrat of Maryland.