WASHINGTON — The Senate voted Thursday to reinvent the nation’s health care system, passing a bill to guarantee access to health insurance for tens of millions of Americans and to rein in health costs as proposed by President Obama.

Vice President Joseph R. Biden Jr. arrived for the vote at the U.S. Capitol on Thursday.

The 60-to-39 party-line vote, on the 25th straight day of debate on the legislation, brings Democrats a step closer to a goal they have pursued for decades. It clears the way for negotiations with the House, which passed a broadly similar bill last month by a vote of 220 to 215.

When the roll was called at 7:05 a.m. on Thursday, it was a solemn moment. Senators called out “aye” or “no.” Senator...