WASHINGTON — The House gave final approval on Thursday to legislation to keep government financed through September, and it also passed a Republican blueprint that enshrined the party’s vision of a balanced budget that would substantially shrink spending, privatize Medicare and rewrite the tax code to make it simpler.

With a final flurry, Republican leaders sent the House home for a two-week recess, confident that they had outmaneuvered President Obama and the Democrats in the running fiscal fight from the last redoubt of Republican control in Washington. In the Senate, Republicans put Democrats on notice that passage of the Senate’s first budget since 2009 would come at a political price.

Democrats had to vote down an amendment, 46-53, on Thursday night that instructed them to rewrite their budget to balance it, something it never does. But in a nonbinding provision, the Senate voted overwhelmingly, 79-20, to repeal a 2.3 percent tax on medical devices that helps finance the president’s health care overhaul, the first step in what Republicans see as a chipping away of the law.

In a marathon session scheduled for Friday, Senate Democrats expect to vote on nonbinding amendments that are tailor-made for the 2014 campaign: to uphold gun rights being challenged in the United Nations; to make the president and vice president buy insurance through the new Affordable Care Act; and to withhold the president’s salary until he submits a budget, as legally required, among dozens of other proposals.