WASHINGTON — Congress on Thursday gave final approval to a package of changes to the Democrats’ sweeping health care overhaul, capping a bitter partisan battle over the most far-reaching social legislation in nearly half a century.

The bill, which Democratic leaders hailed as a landmark achievement, now goes to President Obama for his signature.

“The American people have waited for this moment for a century,” the Senate majority leader, Harry Reid of Nevada, said at a news conference. “This, of course, was a health bill. But it is also a jobs bill, an economic recovery bill, was a deficit-reduction bill, was an antidiscrimination bill. It was truly a bill of rights. And now it is the law of the land.”

In a fitting finale to the yearlong health care saga, the budget reconciliation measure that included the final changes was approved first by the Senate and then by the House on a tumultuous day at the Capitol, as lawmakers raced to complete their work ahead of a two-week recess.