WASHINGTON  Senate Democrats said Saturday that they had clinched an agreement on a far-reaching overhaul of the nation’s health care system and forged ahead with efforts to approve the legislation by Christmas over Republican opposition.

As the Senate convened in a blizzard, Democratic leaders hailed a breakthrough that came when Senator Ben Nelson, Democrat of Nebraska, agreed to back the bill after 13 hours of negotiations on Friday, making him the pivotal 60th vote for a measure that President Obama has called his top domestic priority.

“Change is never easy, but change is what’s necessary in America,” Mr. Nelson said at a morning news conference. “And that’s why I intend to vote,” he said, “for health care reform.”

Mr. Obama, appearing on television from the White House, said: “Today is a major step forward for the American people. After nearly a century-long struggle, we are on the cusp of making health care reform a reality in the United States of America.”